YANG Wei-tao, Clive Woodley, YAN Guang-hu, HU Rui, XIAO Xia

(1.Xi′an Modern Chemistry Research Institute, Xi′an 710065, China; 2.Imperial College London, London, UK)

Abstract:The combustion model of multi-perforated disk (MPD) gun propellant was established to simulate and improve its combustion behaviors. Under the precondition of the geometry combustion law, the calculations describing pressure history and dynamic vivacity relationship of MPD were carried out. Influencing factors such as disk thickness, perforation diameter, burning rate coefficient and exponent were analysed. The results showed that the geometry and formulation influenced the burning characteristics of MPD. The maximum dynamic vivacity and burning time increase with the increase of disk thickness and the decrease of perforation diameter. But the influence of MPD geometry has lower impact on the combustion behavior. Comparatively, the burning coefficient and pressure exponent influence the burning behavior most, and the best propellant formulation should possess higher coefficient and pressure exponent.

Keywords:physical chemistry; multi-perforated disk (MPD); gun propellant; calculated combustion behavior; closed bomb

Introduction

Usually there are two ways to improve the internal ballistic performance in guns include or artillery. The first one is increasing the total energy available in the propulsion system, and the second method is tailoring the release of this energy, temporally and spatially, to maximize its transfer to the projectile[1].

For method one, people devoted themselves to increase the force content of propellant and loading density charge. The force content of the propellant can be varied from 900J/g to 1400J/g from single-base gun propellant to triple-base gun propellant with high energy solid content[2-4]. Meanwhile, the loading density of the charge can be improved from 0.8g/cm3to 1.3g/cm3using consolidated methods, such as consolidated/compressed charges, compact modified propellant charge, monolithic propellant grain and also monolithic foamed propellant[5-7]. Nevertheless, the development of high energy propellant or high energy density charge is limited to the bearing capacity of the gun barrel.

For method two, the release of energy can be adjusted by tailoring the burning rate or the burning surface area. Usually coating, deterring or co-extruding methods are used to adjust the burning rate of each layer[8-11]. Multi-perforated propellant grains with higher combustion progressivity were also fabricated, such as 7-perf, 19-perf 37-perf, programmed-splitting sticks (PSS) propellant grains or sticks[12]. In addition, foamed propellant presented high burning progressivity by filtration combustion[13-14]. Consolidated or compressed charges can also improve the progressivity by splitting during the ignition process[7].

With the development of advanced manufacturing technologies, propellant is not restricted to simple geometries. Complex geometries made using new methods such as 3D printing or casting technology can provide high burning area progressivity, and thus improve the energy efficiency[15-16]. The calculation of combustion behaviors provides a basis for understanding the performance of MPD, benefitting for substantial reduction in time and cost. Thus, the calculation method and design principle for MPD are discussed in this study.

1 Method of calculating surface area and volume

The Figure 1 below shows the cross-section of MPD being considered. It comprises a central hole with hundreds of small perforations evenly distributed over the rest of the area of cross-section. The disk outer diameter (D) is 40mm，central hole diameter (C) is 8mm，the web size (w) is 1mm, and these values will not be changed because the bore diameter of the gun and center core diameter of igniter are constant.

Fig.1 Cross-section of MPD

The following dimension are defined.

D, disk outer diameter;C, central hole diameter;d, perforation diameter;w, web (distance between adjacent perforation);L, length of disk;X, distance burned.

The first step required is to estimate the number of perforations. This process is carried out by estimating the number of inner perforations and the number of outer perforations. The outer perforations essentially are those perforations nearest the outer circumference of the disk and those nearest the inner circumference of the disk.

For the inner perforations, the effective disk cross-section area is divided by the area of a triangle formed by 3 holes as shown below.

Noting that the effective cross-section area of the disk for this purpose is the grain diameter less 2 times the web and minus the perforation diameter.

(1)

An implicit assumption for the MPD grain geometry is that there are many perforations so that the distance between the centers of outer perforations is an approximation to an arc of the grain circumference. Consequently, considering 2 adjacent holes at the outer edge of the MPD, then the following estimate is made for the number of outer holes.

(2)

Taking into account the central hole, the total number of perforation is then the sum of the two equations multiplied by 0.5. The factor of 0.5 is due to the fact that each triangle containing 1/6 of each perforation and each hole on the circumference is only half a hole.

The initial area of cross-section and total surface area of the grain can then be calculated, together with the initial volume, using the equation

Vi=Initial volume=

(3)

Si=Initial surface area=

πL[D+C+2d(ni+n0)]

(4)

After the MPD has started to burn, then the instantaneous surface area and volume are calculated in the normal manner for perforated propellant graining, remembering that the perforation diameter are increased by 2X, the disk diameter is reduced by 2X, the central hole diameter is increased by 2X, the web is decreased by 2Xand the grain length is decreased by 2X.

The area of each triangle formed by 3 inner holes is calculated in an identical way to that calculated for 7-perf, 19-perf and 37-perf propellants, including the slivering process[18-19].

The area formed by the outer hole is approximately by treating the outer edge as a straight line. This approximation is valid if the number of perforation is large. Making this assumption, means that the area and hence volume are calculated in exactly the same way as those for the outer region of hexagonal propellant grains[18-19].

2 Calculation of pressure-time curves in a closed bomb

The linear burning rateriis given by

ri=u1pn

(5)

The mass fraction burning rate of the propellant is

(6)

whereriandSiare the instantaneous value of the burning rate and surface area, respectively, andVgiis the initial grain volume.

Thus, the fraction of mass burned of the propellant is

(7)

The space-mean pressurep(Noble-Abel Law) is

(8)

whereFi,CiandToiare the force content of propellant, initial mass and adiabatic flame temperature of propellant, respectively;FI,CIandToIare the force content of propellant, initial mass and adiabatic flame temperature of igniter propellant, respectively;Tis the mean temperature of propellant gases;Vcis the volume available for gases. The details about the calculation process and equations are described in reference[17-18].

Assuming the igniter is fully burned at zero time, the volume available for gases is shown below.

(9)

Thus, the pressure history can be derived from the above equations, and dynamic vivacity curves can also obtained by the equations below

(10)

3 The influence of geometry on the combustion behavior

The characteristics of the gun propellant are constant and stated below. The adiabatic flame temperature (Tv) is 3584K, force contentfis 1194J/g, density of propellant is 1.6g/cm3, specific heat ratio(γ) is 1.25, coefficient of burning (u1) is 0.06, pressure exponent (n) is 1. The MPD was burned at a loading density of 0.2g/cm3using 1.3g black powder as an igniter.

3.1 Influence of disk thickness (L)

To estimate the influence of disk thickness on the burning characteristics of MPD in a closed bomb, different disks with thickness from 0.25cm to 2cm were modelled. Figure 2 presents thep—tand dynamic vivacity curves of the MPD compared with 7-perf grain with perforation diameter (d) of 0.5mm.

Fig.2 The p—t and dynamic vivacity curves of MPD with different thickness

As Figure 2 (a) indicates, the burning time decreases with the decrease of disk thickness. For a fixed loading density (i.e. a fixed propellant mass), there are more disks and more surface area, resulting in a faster combustion. Meanwhile, Figure 2 (b) demonstrates that the combustion progressivity is better than that of 7-perf propellant grain when the MPD thickness is greater than 0.5cm. Finally, the progressivity of MPD with 0.25cm thickness is poor because the value of B at maximum dynamic vivacity is only 0.4.

Figure 3 presents the maximum dynamic vivacity and burning time of MPD with different disk thicknesses.

Fig.3 Maximum dynamic vivacity and burning time variation with L values

As shown in Figure 3, the maximum dynamic vivacity and burning time present the same variation tendency. When the disk thickness is more than 1cm, the slope of increasing curves becomes smaller.

3.2 Influence of perforation diameter (d)

To estimate the influence of perforation diameterd, different disks with perforation diameter from 0.3mm to 2.0mm were modelled and compared. Figure 4 presents thep—tand dynamic vivacity curves of the MPD with a constant disk thickness (L) of 2cm.

Fig.4 The p—t and dynamic vivacity curves of MPD with different perforation diameters

As Figure 4 shows, the maximum dynamic vivacity increased from 0.0011 to 0.0017MPa-1·ms-1with the decrease of perforation diameter from 2mm to 0.3mm. As Figure 5 show, the burning time decreases when the perforation diameter varies from 0.3mm to 1.0mm. But, the burning time of MPD with 1.0mm and 2.0mm perforation diameter is similar.

Fig.5 Maximum dynamic vivacity and burning time variation with d values

4 Influence of propellant formulations

The adiabatic temperature (Tv) is 3584K, force contentfis 1194J/g, density of propellant is 1.6g/cm3, specific heat ratio (γ) is 1.25. The MPD was burned at a loading density of 0.2g/cm3using 1.3g black powder as an igniter. At the same time, keeping the disk thickness of 2cm, the perforation diameter of 0.5mm constant, the influence of the burning coefficient (u1) and pressure exponent (n) determined by propellant formulation are discussed in this part.

4.1 Burning rate coefficient (u1)

Figure 6 showsp—tand dynamic vivacity curves of MPD with three burning coefficient (u1) values. The burning coefficient varied from 0.03 to 0.1(cm·s-1·MPa-1). As shown in Figure 6, the burning time decreases with the increase of burning coefficient. Also, the maximum dynamic vivacity increased almost 5 times whenu1value decreases from 0.1 to 0.03(cm·s-1·MPa-1), indicating high dependence of combustion progressivity on burning coefficient.

Fig.6 The p—t and dynamic vivacity curves of MPD with burning coefficient (u1)

4.2 Influence of pressure exponent n

Keepingu1=0.06(cm·s-1·MPa-1) constant, the influence of pressure exponent n is discussed here. Three pressure exponent values (1, 0.9 and 0.8) were chosen for calculation. As Figure 7 shows, when the pressure exponent decreased, the maximum dynamic decreased from 0.0015 to 0.0004MPa-1·ms-1. Furthermore, the profile is flat and much similar to that of a propellant with tube geometry, i.e. a neutral surface area function.

Fig.7 The p—t and dynamic vivacity curves of MPD with pressure exponent (u1)

5 Conclusions

Geometry and formulations will influence the burning characteristics of MPD, and the burning coefficient and pressure exponent will contribute the most. Thus, in order to get a MPD with high combustion progressivity, an effective way is to adjust the formulation of the gun propellant. If the formulation of the gun propellant could not be changed due to special demand, it is better to fabricate a MPD with smaller perforation diameter and bigger disk thickness.