Influence of the Ignition Method on the Characteristics of Propellant Burning in the Context of the Geometric Burning Model

Abstract

Mathematical models for the simulation of the operation of various projectile-throwing systems are continuously modified, simultaneously with the development of propellants. For these models, properly determined values of energetic ballistic characteristics of propellants are very important. This paper presents an original test and the results of closed vessel comparative investigations. The study analyses specific combustion characteristics of single-base propellants using conventional black powder ignition, plasma ignition, and gas ignition. The results of the study indicate the need to take into account—in the calculation of the burning rate—not only the dynamics of gas pressure changes but also the real (experimental) form function, which differs from the theoretical one resulting from the assumed geometric burning model. The obtained results allow for a critical analysis of the existing methods used for the determination of values of ballistic characteristics of propellants. Further, it will allow for creating pyrostatic test conditions that would reflect the condition of instantaneous ignition of propellant grains over their entire surface in the most realistic way.

1. Introduction

The basis of the theoretical analysis of the operation of projectile-throwing systems is the mathematical model of the internal ballistic cycle of these systems, which includes the energy balance, the equation of gas generation to the postprojectile space, and the equations of mechanics for the motion of the projectile in the barrel [1,2,3]. The equations for this model, formulated in accordance with the nature of the design solution, allow us to describe changes in pressure p of the propellant gases in the postprojectile space of the barrel and velocity v of the propelled projectile (as a function of time t and as a function of path l of the projectile in the barrel) [4,5,6,7,8,9,10].

The accuracy of the model solution is usually determined by input data. Some of these describe the mass-geometric characteristics of the projectile-throwing system elements and can be set with satisfactory accuracy. Their values are derived from direct measurements and depend only on the known accuracy of the measuring instrument. The energy characteristics of solid propellants (e.g., force and covolume) can also be determined easily. In addition, their values are derived from measurements of propellant gas pressure, charge weight, and propellant density. The results of experimental pyrostatic tests (closed vessel tests) enable us to validate the values of the force and covolume of the propellant gases, in terms of the thermochemical calculations. Furthermore, thermochemical calculations also allow us to determine the temperature and adiabatic exponents of the propellant gases for a known chemical composition of a propellant [11,12,13,14]. It is much more difficult to achieve such accuracy for the ballistic characteristics, i.e., the relationship between the burning rate and the pressure of propellant gases surrounding the burning surface of the propellant grains, along with the initial value of the propellant temperature. The values of ballistic characteristics depend on both direct measurements (pressure) and the use of a geometric burning model. Unfortunately, the assumptions of this model (in particular, the uniform shape and dimensions of propellant grains, and simultaneous ignition of grains) turn out to be unrealistic, especially for fine-grained propellants, which generally leads to inconsistencies between the results of theoretical calculations and experimental results. An understanding of these relationships, known as the pressure function and the temperature function of the burning rate law in internal ballistics, enables the modelling of the inflow of propellant gases to the postprojectile space in the barrel. For this purpose, theoretical analyses and experimental studies involving the processes of ignition and combustion of propellants are used.

Combustion process models are used for the theoretical determination of such relationships. The foundations of solid fuel combustion theory (including solid propellants) were established by Zeldowich in the 1940s [15]. Researchers have continued to develop this theory, which is reflected in monographs [16,17,18,19,20,21], review articles [22,23,24], and reports [25]. Ignition models, based on the detailed kinetics of chemical reactions, are a useful tool for analysing the ignition process. However, they are too complicated, so simple ignition criteria are applied in internal ballistics models. The most common criterion is the specific temperature value on the propellant grain surface, called the ignition temperature [26,27,28,29,30,31].

Experimental pyrostatic tests are also used for determining the pressure and temperature functions of the burning rate law. They are based on burning a specific propellant mass in a closed volume and recording the changes in the pressure of the gases from the combustion of this mass (over time). However, the way in which the ignition is carried out or the heterogeneity of the shape and dimensions of the propellant grains can affect the experimental results.

Traditional materials which initiate the ignition of propellants are black powder, boron potassium nitrate (BKNO₃), a mixture of nitrocellulose and black powder (Benite) or an igniter, in which about 98% consists of nitrocellulose, also known as a clean-burning igniter (CBI). These materials initiate ignition by combining convection heat transfer from the hot gases and the heat conduction through the hot particles of the ignition material deposited on the surface of the propellant grains. However, it should be noted that the conditions of pyrostatic testing differ significantly from the conditions of propellant burning in existing projectile-throwing systems. In addition, the ignition systems and conditions used during pyrostatic tests are based on standardized pyrostatic test methods, which differ considerably from those used in modern projectile-throwing systems. In [32], the ignition material mass is not determined precisely but there is only an indication that it may be black powder, CBI, or Benite, and the ignition system may include an electric match. The possible ignition systems indicated in [33] are an electric match and 0.5–1.0 g of black powder, or a resistance wire spiral and 1 g of CBI. On the other hand, [34] uses black powder as an ignition material during the pyrostatic tests, determining its mass indirectly by the condition of a conventional ignition pressure of 3 MPa, obtained in a closed vessel. The appearance of low-sensitivity propellants forced the search for new methods to ignite this type of propellants during pyrostatic tests, as well as in the designed propulsion systems. Thermite compositions consisting of a metal fuel and an oxidant (e.g., WO₃/Al, Bi₂O₃, and Al/CuO/Fe₃O₄) and foamed nitrocellulose with the addition of titanium particles were tested as alternative ignition materials [35,36,37,38,39]. In the case of other ignition methods, there has been interest in electrothermal-chemical (ETC) ignition—where the factor directly affecting the propellant is hot plasma—laser ignition, and gas ignition (e.g., a methane/oxygen mixture) [40,41,42,43,44,45,46,47,48,49,50,51,52,53].

Compared to classic ignition, the advantage of ETC and gas ignition is that the ignition delays are stabilized at a much lower level, and the plasma pulse covers a much larger surface of propellant grains, which makes this ignition method much closer to the geometric burning model of propellant grain [34,54,55,56], where it is assumed that:

-

There is a chemical and physical homogeneity of the propellant material;

-

All propellant grains are the same, in terms of geometric shape and dimensions;

-

The ignition of all propellant grains is instantaneous and affects their entire available burning surface;

-

The combustion area—according to Piobert’s law—moves in parallel layers into the propellant.

However, the assumptions of the geometric burning model differ significantly from reality, which means that, during the pyrostatic testing carried out in accordance with [32,33], there might be actual reasons for obtaining ambiguous results from tests on the propellant combustion dynamics. A number of publications have been dedicated to this issue, especially in relation to ignition with black powder and the combustion of fine-grained propellants. A critical analysis of the geometric burning model, assuming the same geometric shapes and dimensions for all propellant grains, was conducted in [57,58,59,60,61,62]. The impact of the deformed shape of propellant grains on the possible dispersion of burning rate values was discussed. The assumption that the ignition of all propellant grains is instantaneous and affects the entire available burning surface was critically analysed in [63,64,65]. The possibility of obtaining different values of the dynamic burning characteristics for a given propellant for non-standard ignition conditions and loading density was indicated in [35,66,67,68,69,70,71,72]. The papers above focused mainly on the analysis of the impact of the pressure rise rate as the main factor determining burning rate r(p) and dynamic vivacity L, using the methods described in [32], but still under the assumptions of the geometric combustion model.

This paper indicates the need to take into consideration the experimental form function when determining the burning rate r(p), according to the relationship presented in [32]. The work carried out and presented here involves both the area of experimental research and theoretical analysis. The experimental part consisted of burning a certain mass of single-base propellant under isochoric conditions (closed vessel test) and recording changes (as a function of time) in the pressure of propellant gases. Various materials and ignition systems were used to initiate propellant combustion, such as black powder, methane/oxygen gas mixture, plasma generator, cartridge primer, and electric match. During the experimental tests, the propellant was positioned in the combustion chamber in various ways: the propellant charge was grouped at a specific location on the wall of the combustion chamber, or the propellant charge was located on a paper tape in the form of individual grains, over the entire cylindrical surface of the combustion chamber. In the analytical part, the recorded pressures p(t) of the propellant gases were used both to analyse the dynamics of pressure changes (derivative dp/dt, and relative quickness RQ), and to calculate the experimental form function. The experimental form function is herein compared with the theoretical function. In the final part of the paper, the effect of ignition conditions and the distribution of propellant grains in the combustion chamber on the determination of the burning rate based on the relationship specified in [32], but taking into account the experimental form function, is demonstrated.

Observations and experience based on the presented analyses can be the basis for creating pyrostatic test conditions that would reflect the condition of instantaneous ignition of propellant grains over their entire surface in the most realistic way.

2. Analytical Analyses

The analysis of the problem depends on the following relation for determining the burning rate r [32]:

$$r=\frac{de}{dt}=\frac{de}{dz}\frac{dz}{dp}\frac{dp}{dt}$$

(1)

for which the input data are the following terms:

-

Term de/dz, resulting from the analysis of changes in the thickness of the burning layer of propellant grains:

$$\frac{de}{dz}=\frac{V_{g0}}{S_{g0}·\varphi \left(z\right)}$$

(2)

where:

e—the thickness of the burning layer of the propellant grain;

z—the relative burned mass of propellant;

ϕ—the relative burning surface;

S_g0 and V_g0—the initial surface and initial volume of propellant grains respectively.

Equation (2) follows from the mass balance equation in a closed volume:

$$\frac{dz}{dt}=\frac{S_{g0}}{V_{g0}}·\frac{S_g}{S_{g0}}·\frac{de}{dt}=\frac{S_{g0}}{V_{g0}}·\varphi (z)·\frac{de}{dt}$$

(3)

For the case when propellant burns in a closed volume without mass exchange with the surroundings, the mass of gases resulting from combustion will always be equal to the mass of the burned part of the propellant charge.

-

Term dz/dp, resulting from the transformation of the basic pyrostatic equation

$$p=\frac{f·m_p·z}{W_0-\frac{m_p}{\rho }\left(1-z\right)-\eta ·m_p·z}$$

(4)

to the following form:

$$\frac{dz}{dp}=\frac{f·\left(\frac{1}{∆}-\frac{1}{\rho }\right)}{{\left[f+\left(\eta -\frac{1}{\rho }\right)·p\right]}^2}$$

(5)

where:

f—the force;

η—the covolume;

Δ—the loading density;

ρ—the propellant density.

-

Term dp/dt, resulting from the experimental curve p(t).

The input data for Equation (5) in the form of values of force f, covolume η, propellant density ρ, loading density Δ, and current pressure p, as well as the input data for Equation (2) in the form of initial area S_g0 and volume V_g0 of propellant grains can be determined with intended accuracy.

More attention should be paid to relation ϕ(z), described in (2) as the relative change of the surface of propellant grains during their combustion and pressure derivatives dp/dt. The assumptions of the geometric combustion model for propellant grains require the use of theoretical relations ϕ(z) arising from the specific shape of the grains [1,2,3,34]. As a result of the analyses, it will be demonstrated that the experimental form function ϕ_ex(z), differs from the theoretical function.

Deviations from the geometric burning model may be a result of the dispersion of the shapes and sizes of propellant grains, non-simultaneous ignition or deformation, and cracking of grains. For cases with large deviations from the geometric burning model in the combustion process, the authors propose an analysis method based on the idea of an experimental form function [64]. The equation for the inflow of propellant gases can be written as:

$$\frac{dz}{dt}=\frac{S_{g0}}{V_{g0}}·r·{\varphi }_{ex}(z)$$

(6)

Taking the power form of function r(p) in the form of the following expression:

$$r=\beta {\left(\frac{p}{p_0}\right)}^{\alpha }, p_0=0.1 \mathrm{M}\mathrm{P}\mathrm{a}$$

(7)

resulting in the equation

$$\frac{dz}{dt}=\frac{S_{g0}}{V_{g0}}·\beta ·{\varphi }_{ex}(z){\left(\frac{p}{p_0}\right)}^{\alpha }$$

(8)

which can be written as follows:

$$\frac{dz}{dt}=G(z)·{\xi }^{\alpha }$$

(9)

where

$$G\left(z\right)=\theta ·{\varphi }_{ex}\left(z\right),\hspace{1em}\hspace{1em}\hspace{1em}\theta =\frac{S_{g0}}{V_{g0}}·\beta ,\hspace{1em}\hspace{1em}\hspace{1em}\xi =\frac{p}{p_0}$$

The above form of the gas inflow equation, in the case of α = 1, is analogous to the so-called ‘physical burning law’ introduced in the monograph [34]. In this case, function G(z) is analogous to function Γ(z), introduced in [34]

$$G\left(z\right)=\Gamma \left(z\right)=\frac{S_{g0}}{V_{g0}}·\frac{S_g}{S_{g0}}·r_1=\frac{S_{g0}}{V_{g0}}·\varphi (z)·r_1$$

(10)

where r₁-coefficient of the linear form of the burning rate law (r = r₁ · p).

Equation (9), for α ≠ 1, is a generalization of the physical burning law to the power-law relationship between burning velocity and pressure. Both sides of (9) can be expressed as logarithms:

$$\lg \left(\frac{d z}{d t}\right)=\lg \left[G\left(z\right)\right]+\alpha \lg \xi$$

(11)

For a fixed value z, the above equality represents a linear relationship between lg(dz/dt) and lgξ. Therefore, by calculating lg(dz/dt) for several values of loading density and approximating the linear function of the relation between these values and lgξ we can determine the value of exponent α in the burning rate law. After calculating value α, function G(z) is determined by:

$$G\left(z\right)=\frac{d z}{d t}{\xi }^{-\alpha }$$

(12)

As shown in (9), there is a simple relation between function G(z) and the experimental form function ϕ_ex(z):

$${\varphi }_{ex}\left(z\right)=\frac{G\left(z\right)}{\theta }=\frac{G\left(z\right)}{\frac{S_{g0}}{V_{g0}}·\beta }$$

(13)

These functions only differ in the constant multiplier θ. It can be determined by establishing the value of coefficient β based on the relation (10):

$$\beta =\frac{e_{gb}\left(z_2\right)-e_{gb}\left(z_1\right)}{{\displaystyle \underset{z_1}{\overset{z_2}{\int }}{G}^{-1}\left(z\right)dz}}$$

(14)

where e_gb—burned layer of grain, and values e_gb correspond to values z₁ and z₂.

As mentioned earlier, one of the reasons that can quantitatively affect the determination of the burning rate value according to relation (1) can be the pressure derivative dp/dt calculated from the experimental curve p(t). It will be shown later in this paper that the dynamics of the change in p(t) depend on the method used to ignite the propellant.

This can be demonstrated by relative quickness (RQ). This value expresses the reference value of the pressure rise rate between the tested propellant and the reference propellant as a percentage. The reference propellant is defined as a propellant whose ignition is initiated under the conditions specified in STANAG 4115 [32], while the tested propellant is a propellant whose ignition conditions deviate significantly from those specified in [32].

RQ was introduced in [33] and its value for a single test sample is calculated on the basis of the following relation:

$$RQ=\frac{100}{4}\sum _{j=1}^n\frac{T_j}{R_j}$$

(15)

where:

T_j—value of dp/dt for the tested propellant at the j-th point;

R_j—value of dp/dt for the reference propellant at the j-th point.

As a rule, four test points on the curve dp/dt = f(p) are chosen, corresponding to 27, 40, 53, and 66% of the maximum pressure value.

3. Experimental Procedure and Materials

To solve the problem, experimental investigations of single-base propellants were conducted in closed vessels applying different ignition methods. Closed vessel investigations were carried out in two different vessels: a conventional closed vessel (CCV) of 200 cm³ (under loading densities of 100 and 200 kg/m³) and a micro-closed vessel (MCV) of 1.786 cm³ (under loading densities of 100 kg/m³). All tests were conducted at ambient temperature.

The pressure was measured with an HPI 5QP 6000M piezoelectric transducer, whose signal was amplified by a TA-3/D amplifier and recorded on a Keithley DAS-50 12-bit analogue-to-digital converter at a frequency of 1 MHz. Pressure courses were sampled with a time step equal to 25 μs. The maximum systematic error of the pressure indirect measurement system was 1.1% [49,58,69,70,71]. The recorded courses of pressure p of the propellant gases, as a function of time t, were the bases for further analyses.

The tested materials were a single-base seven-perforated propellant (propellant A) and a single-base single-perforated propellant (propellant B). The average geometrical dimensions of propellant grains A and B are shown in

Table 1. Average dimensions of grains of single-base propellants A and B.

Shape	Average Dimensions of Grain (mm)
	Total Layer (1/2 of Web Size)	Inside Diameter	Length
seven-perforated (A)	0.2550	0.18	3.4
single-perforated (B)	0.1625	0.15	1.9

.

Table 2. Energy characteristics for propellants A and B.

Propellant	Energy Characteristics of Propellant
	Force f (J/kg)	Covolume η (m3/kg)
A	903,000	0.00149
B	1,023,000	0.00115

summarizes the values of the energy characteristics (force f and covolume η) of propellants A and B. The values of energy characteristics will be used to determine the dz/dp component in Equation (5).

3.1. Detailing the Experimental Procedure for Propellant A

Pyrostatic testing of propellant A, enabling the analysis of the experimental form function ϕ_ex(z) and pressure derivative dp/dt, was conducted in a conventional closed vessel (CCV) for the following conditions of ignition and distribution of propellant A in a closed vessel:

(1)

Conventional black powder ignition (Figure 1); the tested propellant positioned on the combustion chamber wall (as described in STANAG 4115) or evenly distributed on a paper tape over the entire internal surface of the combustion chamber wall (Figure 2);

(2)

Ignition using a mixture of gases (oxygen, methane) introduced to the combustion chamber (see test stand in Figure 3); the tested propellant positioned on the combustion chamber wall (as described in STANAG 4115), or evenly distributed on a paper tape over the entire internal surface of the combustion chamber wall (Figure 2);

(3)

Ignition by a plasma pulse generated in a capillary plasma generator (CPG—Figure 4); the tested propellant positioned on the combustion chamber wall (as described in STANAG 4115).

Figure 1. Ignition plug equipped with black powder igniter.

Figure 1. Ignition plug equipped with black powder igniter.

Figure 2. Propellant grains spaced separately and systematically on the whole circuit of the combustion chamber.

Figure 2. Propellant grains spaced separately and systematically on the whole circuit of the combustion chamber.

Figure 3. Test stand for gas ignition investigation: closed vessel on the left-hand side, gas tank on the right-hand side.

Figure 3. Test stand for gas ignition investigation: closed vessel on the left-hand side, gas tank on the right-hand side.

Figure 4. Ignition plug equipped with plasma generator: (a) cross-section and (b) real construction.

Figure 4. Ignition plug equipped with plasma generator: (a) cross-section and (b) real construction.

Investigation of propellant A was carried out for a density of 100 kg/m³ only if the propellant grains were located on a paper tape. Only for such a loading density was it possible to locate in the combustion chamber a paper tape on which all the grains of a certain mass of propellant were located.

For any other (larger) mass of propellant, this was impossible due to the limited area of the tape corresponding to the cylindrical surface of the combustion chamber.

3.2. Detailing the Experimental Procedure for Propellant B

Pyrostatic testing of propellant B, enabling the analysis of experimental form function ϕ_ex(z) and pressure derivative dp/dt, was conducted in a conventional closed vessel (CCV), and a micro closed vessel (MCV), as described in [69]. The tests were conducted for the following conditions of ignition and distribution of propellant B in the closed vessel:

(1)

Conventional black powder ignition (Figure 1), in which different masses of black powder were used, with the tested propellant positioned on the wall of the combustion chamber (as described in STANAG 4115);

(2)

Conventional black powder ignition, with the tested propellant and black powder igniter placed in a bag made of flammable material;

(3)

Ignition using only an electric match;

(4)

Ignition of the tested propellant in the micro closed vessel (MCV) using a cartridge primer.

4. Results of Pyrostatic Tests and Their Analysis

4.1. Pressure Rise Rate

4.1.1. Propellant A (Black Powder Ignition and Gas Ignition)

In gas ignition, the initial pressure of the methane/oxygen mixture was changed in the range 0.15–0.25 MPa. The mixture ignition was initiated using a standard electric match. An increase in the gas mixture pressure causes a rise in its mass and, thus, the energy released during combustion. This accelerates the ignition process. The pressure rises very quickly and then either stays relatively constant or rises much more slowly (Figure 5 and Figure 6). Then, the pressure begins to increase due to the ignition of the tested propellant. This increase is faster than in the case of black powder ignition.

Figure 7 shows full pressure graphs recorded with gas ignition for initial gas mixture pressure values of 0.15, 0.20, and 0.25 MPa (propellant placed loosely in the chamber). The pressure graph recorded using black powder ignition is also shown, for comparison. Analogous graphs for a propellant attached to a tape are shown in Figure 8. It can be seen that propellant burns faster when it is placed on the tape.

In the case of gas ignition, the pressure rise is faster than in black powder ignition. To illustrate this, Figure 9 and Figure 10 compare the time plots of the pressure derivative. In the black powder ignition, the propellant distribution has a visible effect on the pressure derivative (Figure 9). Figure 10 shows the pressure derivative graphs made using the gas ignition for various propellant distributions. As can be seen, the increase in ignition energy causes an increase in the maximum derivative value; in addition, there is a clear impact of the propellant distribution on the pressure derivative graphs. As the ignition energy increases, this effect becomes weaker. This effect can be explained by the easier access of hot gases to the surface of the propellant grains when the propellant is distributed on the walls of the chamber. This demonstrates the important role of the spatial location of the igniter and propellant in determining the characteristics of the propellants, on the basis of the test results in the closed vessel.

These observations are confirmed by the relative quickness values (

Table 3. RQ value (in %) for black powder and gas ignitions.

Tested Propellant in the Closed Vessel	Ignition Method
	Black Powder	pM/O 0.15 MPa	pM/O 0.20 MPa	pM/O 0.25 MPa
loosely	100.00	94.70	104.47	119.72
on tape	103.43	105.38	110.60	135.14

). The black powder ignition of propellant A, distributed loosely in the closed vessel, was treated as a reference test.

4.1.2. Propellant B (Classic Ignition with Black Powder of Various Masses and Ignition Using a Cartridge Primer)

The analysis of the gas pressure rise rate during the pyrostatic testing of propellant B is shown in Figure 11, Figure 12, Figure 13 and Figure 14. Figure 11 and Figure 12 present the ignition of propellant B with black powder weighing 2 g and using only an electric match. For black powder ignition, cases were compared when the tested propellant grains were loose on the wall of the combustion chamber (according to STANAG 4115) and when the black powder and the tested propellant were together in a bag. The situation where the black powder and the tested propellant were together in a bag contributed to greater dynamics in the burning process.

Figure 13 and Figure 14 show graphs of pressure p(t) and pressure derivative dp/dt for the ignition of propellant B with black powder weighing 2, 4, 6, and 8 g (in the CCV) and as a result of activated cartridge primer (in the MCV). The increase in the mass of black powder significantly reduced the burning time of propellant B. The increase in the mass of black powder caused a rise in the energy released from combustion, which accelerated the ignition process.

Ignition with a cartridge primer was characterized by much higher combustion dynamics for propellant B, compared to black powder ignition. These observations were confirmed by the relative quickness values presented in

Table 4. RQ values (in %) for ignition with black powder, an initiating head, and cartridge primer.

Ignition Type
Electric Match	2 g of Black Powder	2 g of Black Powder	4 g of Black Powder	6 g of Black Powder	8 g of Black Powder	Cartridge Primer
91.82	100.00	114.37	111.05	116.35	126.73	130.81
propellant placed loosely in the combustion chamber		propellant in a bag with black powder	propellant placed loosely in the combustion chamber

. Black powder ignition of propellant B spread loosely in the closed vessel and weighing 2 g was treated as a reference test.

4.2. Experimental Form Function

4.2.1. Propellant A (Black Powder Ignition, Gas Ignition, and Plasma Ignition)

The analysis of form function variation of propellant A is presented in Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20. Figure 15, Figure 17 and Figure 19 present the graphs of the experimental form function for using gas ignition while Figure 16, Figure 18 and Figure 20 present the graphs of the experimental form function for black powder ignition. The figures also show the graphs of the theoretical form function determined using the geometric burning model [34]. Two experimental form function curves in Figure 16, Figure 18 and Figure 20 result from the analysis of two test samples of a given propellant (conducted under the same conditions). They illustrate the repeatability of the experimental results.

Graphs of the experimental form function, for gas ignition at a gas mixture pressure of 0.15 MPa, show the fastest increase with value z. This indicates that, in this case, the ignition is close to simultaneous ignition. This is the result of a long ignition time, in which the surfaces of the grains in a large part of the charge have time to heat up to the ignition temperature. The greater the mixture pressure is, the more the ignition deviates from simultaneous ignition. This can be explained as follows: A portion of the propellant charge ignites quickly. This may be due to the intense movement of the combustion products in the gas mixture. Oscillations on the rising part of curves ϕ_ex(z) are a reflection of this movement. The greater the initial pressure of the mixture, the more intense these oscillations are. Faster ignition of part of the charge results in a faster pressure increase than at lower values of the gas mixture pressure. Therefore, part of the propellant charge is burnt, to a large extent, before the rest of it ignites.

In the graphs of the experimental form function at the loading density of 100 kg/m³ and initial values of the mixture pressure of 0.15 and 0.20 MPa, function ϕ_ex(z) first increases and then temporarily decreases. This effect can be explained by the heating of the external parts of the grains. As a result, the flame propagates in the heated material. The speed of the flame propagation is higher because of temperature effects.

As the heated layer burns, the burning rate decreases. This causes a drop in the value of ϕ_ex(z). For the initial value of the gas mixture pressure of 0.25 MPa, a part of the propellant charge is ignited rapidly. The generated propellant gases ignite the rest of the charge. The resulting increase in the burning surface outweighs the decrease in the burning rate, due to the burning out of the heated layer of propellant grains. Thus, there is no decrease in the value of ϕ_ex(z). In the curves for a loading density of 200 kg/m³, ϕ_ex(z) did not have the characteristics observed at a loading density of 100 kg/m³ because there was twice the mass of the propellant charge and the resulting smaller thickness of the heated external layer of propellant grains.

In the case of black powder ignition, some of the propellant grains ignite while the black powder is burning. This is the result of the direct impact of the burning black powder grains on the ignited propellant grains. The remaining part of the propellant charge ignites as the grain surface is heated by the propellant gases. This is a less effective ignition method, as the propellant gases cool down by giving off heat to the propellant grains. By the time the charge is fully ignited, a large part of the propellant has already burned (30–40% of its mass).

Figure 19 and Figure 20 illustrate the impact of the distribution of the propellant charge (L—loose; T—on tape) on the graphs of the experimental form function.

For gas ignition, the impact of the propellant distribution depends on the ignition energy. In the case of the lowest energy, there is an evident acceleration of ignition.

In the first, fast phase of the process, a larger part of the charge is ignited, compared to the loose propellant. Better contact of the propellant with the gas mixture’s combustion products is conducive to the heating of the grains. As a result, the accelerated burning of the external layer of the grains is stronger than in the case of loose propellant. For the gas mixture pressure greater than 0.15 MPa, the impact of charge distribution is clearly weaker. The ignition energy is so high that the propellant charge distribution does not matter.

In the case of black powder ignition, the way the propellant is distributed in the chamber affects the initial part of the graphs, in terms of ϕ_ex(z). The burning black powder particles have better access to the propellant grains attached to the tape. Because these particles fall to the bottom wall of the chamber (the chamber is placed horizontally), it is primarily the propellant grains on this wall that ignite. The remaining grains are ignited by the propellant gases generated when the propellant is burning.

Therefore, parts of the graphs ϕ_ex(z) for z > 0.2 practically coincide with the graphs for the loose propellant in the chamber.

Figure 21 and Figure 22 show the graphs of the experimental form function determined for three ignition methods: black powder ignition (GP), gas ignition (GA), and plasma ignition (PL). The presented form functions correspond to similar values of ignition energy. Gas ignition and plasma ignition accelerate the propellant ignition. For plasma ignition, there are dynamic effects related to the fast movement of the plasma stream.

4.2.2. Propellant B (Black Powder Ignition, Different Distribution of Ignition Charge and Main Charge)

Figure 23 shows the graphs of the experimental form function for different values of black powder mass. The ‘Z’ graph refers to the ignition using an electric match only (black powder mass of 0). Figure 24 illustrates the effect of the relative distribution of the igniting charge and ignited propellant: (L) loose ignited propellant, (B) black powder and ignited propellant in one bag, and (P) propellant ignited with a cartridge primer.

As the mass of black powder increases to 6 g, the degree of ignition of the propellant rises significantly at the moment when the ignition pressure is reached. This means higher values of ϕ_ex for z = 0. This is the effect of delivering more energy to the ignited propellant; however, there is a saturation effect here. Increasing the mass of black powder to 8 g does not practically increase the degree of ignition.

It can be assumed that, with the black powder mass of 6 g, the entire surface of the propellant bed available for ignited black powder particles has been used. Another possibility is an incomplete ignition of the black powder charge with a large charge mass. Some of the black powder grains only ignite because of interactions with the combustion products of the tested propellant.

Attention should be paid to the fact that the graphs of the experimental form function become closer to the theoretical graph as the black powder mass increases. Another noteworthy feature of graphs ϕ_ex(z) is their proximity to z > 0.4. The clearly degressive nature of combustion for this range of z can be explained by the delayed ignition of propellant inside the perforation. The increase in the burning surface inside the perforation does not compensate for the decrease in the burning surface on the external parts of the grains. It can be assumed that the igniter only ignites the external surfaces of grains. The surface inside the perforation ignites with a certain delay after the ignition of the external surface of grains.

The similarity of graphs ϕ_ex(z) for z > 0.4 can also be seen in the graphs presented in Figure 24, despite the different nature of the graphs for z < 0.4. Placing the black powder and tested propellant in a bag (B) was particularly advantageous because it ensures good contact between the propellant and the burning black powder. As a result, graph ϕ_ex(z) is close to the theoretical graph for a wide range of z values [0.1; 0.6]. Graph ϕ_ex(z) for cartridge primer ignition reflects the dynamic nature of the process. In accordance with the results of the experimental tests [73] and the analysis carried out in [74], the pressure value inside the cartridge primer is about 100 MPa. A large difference in the pressure values inside the cartridge primer and inside the chamber causes an intensive outflow of combustion products of the ignition mass through the cartridge primer channel. This outflow generates a shock wave in the air filling the chamber (the propellant bed only occupied about 13% of the chamber volume).

The shock wave reaches the wall closing the chamber and is reflected; the multiple reverberations affect the course of the recorded pressure. The intense movements of the ignition mixture’s combustion products facilitate rapid ignition, and this is reflected by the very rapid increase in the pressure recorded for the cartridge primer ignition.

4.3. Burning Rate

The graphs of the experimental form function in Section 4.2 prove the importance of the ignition conditions for the course of propellant burning in tests in the conventional closed vessel. They affect the determined relationships between the burning rate and the pressure, as illustrated by the graphs in Figure 25, Figure 26, Figure 27 and Figure 28.

Based on the presented test results and analysis, the influence of ignition conditions on the determination of the burning rate is clearly evident. For many years, the basis of the method for determining the energy ballistic properties of propellants has been closed vessel testing, for which standard procedures have been established [32,33]. The results of the presented research can be a contribution to the verification of the conditions of closed vessel tests especially in the context of the development of new propellants and projectile-throwing systems containing non-standard ignition systems.

5. Conclusions

The main idea of the research and analysis carried out was to pay attention to the possibility of obtaining different values of the propellant burning rate depending on the conditions for the implementation of closed vessel tests, with particular emphasis on ignition. The main focus was on the analysis of the experimental form function, the determination of which is not strictly dependent on the geometric burning model. The main conclusions of the research and analysis are formulated below:

(1)

The presented analysis of the test results in a conventional closed vessel proves that the determined ballistic properties of propellants significantly depend on the ignition conditions. The RQ values, which characterize the burning rate, may vary over the range of 30–40%. The values of the burning rate determined at different ignition conditions may vary by up to 20%.

(2)

In the case of fine-grained propellants, the use of black powder with a mass sufficient to obtain reliable ignition yields effects which are significantly different from the simultaneous ignition of the tested propellant. To get closer to achieving simultaneous ignition, it is necessary to increase the mass of the ignition charge by several times.

(3)

The use of gas or plasma ignition allows the ignition process to be closer to simultaneous ignition.

(4)

The ignition process can be made closer to simultaneous ignition by changing the propellant distribution in the closed vessel, or by changing the mutual location of the ignition charge and the tested propellant.

(5)

Different methods of bringing the ignition process closer to simultaneous ignition have little impact on the graphs of the experimental form function after fully charged ignition. This shows that non-simultaneous ignition is not the only reason for the deviation of the experimental form function from the function determined by the geometric burning model.

The authors’ future intention is to conduct similar studies for other types of propellants, including double-base and insensitive propellants.