Using the Impulse Method to Determine High-Pressure Dynamic Burning Rate of Solid Propellants

Abstract

A new method for determining the burning rate of a solid propellant, called the Impulse Method, is proposed in this paper. It is based on the proportional relationship between the impulse generated and the mass of the burned propellant. The pressure–time and thrust–time curves are obtained from a tubular propellant grain burning in the chamber, whose inner surface serves as the initial burning surface. Consequently, the mass of the propellant that was burned off at different pressures can be determined, and the burning rates at different pressures are derived according to the geometric parameters of the propellant grain. The Impulse Method was applied to test the burning rate of two types of propellants twice. The results show that the burning rates were consistent for the same propellant at corresponding pressures, demonstrating the feasibility and reliability of the Impulse Method. The burning rate of a GAP-based composite propellant at 20 MPa measured using the Standard Motor Method was 22.6 mm/s, and that measured using the Impulse Method was 22.2 mm/s and 22.7 mm/s, respectively. These findings indicate that the two methods have comparable accuracy. However, the Impulse Method has the advantage of obtaining the burning rate of the solid propellant at any pressure through a single test. In addition, the nozzle erosion only affected the pressure and not the burning rate. Finally, the rationality of the approach for determining the actual specific impulse was proven by comparing the results with those from another testing method.

1. Introduction

The burning rate and pressure exponent of a solid propellant are critical performance parameters for designing solid rocket motors [1]. The combustion mechanism of a solid propellant is complex and influenced by various factors, such as the formulation, pressure, initial temperature, gas flow velocity, and condensed combustion products [2,3]. However, despite extensive research, the current understanding of the underlying mechanisms is still limited. This lack of understanding hinders the accurate prediction of the solid propellant burning rate. Therefore, experimental measurements are generally employed to determine the solid propellant burning rate.

Traditional methods used for determining the burning rate include the Strand Burner Method and the Underwater Acoustic Emission Method [4]. While these methods have been widely used, researchers have continuously strived to improve their accuracy and efficiency [5,6,7]. In recent years, there has been a growing interest in studying the combustion performance of solid propellants under high-pressure conditions [8]. The Closed Bomb Method can be used to test the high-pressure burning rate of propellants. However, it faces challenges in accurately correcting the heat loss and determining the average molecular weight of the combustion gas [9,10,11,12]. To validate steady-state and unsteady-state burning theories of solid propellants, researchers have focused on measuring the transient burning rate. Several techniques have emerged as effective means of testing the transient burning rate of solid propellants, including the Microwave Method [13,14,15,16] and the Ultrasonic Method [17,18,19,20,21,22,23]. Furthermore, the application of laser and photography techniques [24,25,26,27] provides added avenues for measuring the transient burning rate.

One of the notable methodologies proposed in the literature is the Mass Flow Rate Method [28,29]. This method offers a wide pressure testing range and enables the measurement of the dynamic burning rate at a high pressure. However, it relies on the assumption that the nozzle throat diameter and the characteristic velocity remain constant during data processing, which can introduce errors in the results.

In this paper, we propose a novel method for determining the dynamic burning rates of solid propellants at a high pressure based on the fundamental principles of solid rocket motor operation. It utilizes the gas generated from the burning of the propellant itself to create a high-pressure environment, which eliminates the need for pipelines of constant pressure systems and external high-pressure gas sources. Furthermore, the method enables the determination of the burning rate at any desired pressure within an extended pressure range through a single experiment. With this method, we can achieve testing with pressures of up to 40 MPa, allowing for a comprehensive investigation of the propellant burning rate under high-pressure conditions.

2. Measurement Principle and Experiment

2.1. Test System

Figure 1 shows the testing system of the Impulse Method for determining the high-pressure dynamic burning rates of solid propellants. The system consists of five essential components: a compact thick-walled solid rocket test motor equipped with sensors, a vertical test stand, a data acquisition device, an igniter, and a data processing device. The test motor, which is equipped with pressure and thrust sensors, is securely mounted on the vertical test stand. The data acquisition device connects the sensors and the data processing device, allowing for real-time data transfer and analysis. The igniter triggers the primer cartridge in the test motor to initiate the ignition process. And the solid propellant grain under examination is loaded freely in the chamber of the test motor, where it undergoes ignition and subsequent burning.

During the burning process, the integrated testing and data acquisition systems capture and record the thrust produced by the test motor, as well as the pressure signals within the chamber. These signals provide valuable information on the dynamic burning rate of a propellant under high-pressure conditions.

The propellant grain is designed as a tubular configuration, as shown in Figure 2. The end faces and outside surfaces of the propellant grain are coated to prevent burning, leaving only the inner surfaces exposed as the initial ignition surface. During testing, the propellant grain’s initial burning surface is uniformly ignited in perfect synchronization, and the subsequent burning surface follows the parallel layer rule. The propellant exhibits stable burning, without any unstable burning such as erosive burning. Moreover, the actual changing pattern of the burning surface matches the design specifications. In addition, it is assumed that the combustion can respond quickly to the pressure changes.

2.2. Principles

2.2.1. Data Processing

When the primer cartridge in the test motor is ignited, a distinct peak ignition pressure is observed, as indicated by point A in Figure 3. Then, at time t₁, the propellant’s initial burning surface is uniformly ignited, and the propellant burns according to the parallel layer rule. The burning surface of the propellant expands from its initial state, resulting in increased pressure and thrust signals over time. At time t₂, the propellant grain is completely burned out, causing a sharp drop in the chamber pressure and motor thrust. Figure 3 shows the time-dependent variations of the chamber pressure and motor thrust during this process.

In Figure 3, the curves are pressure–time and thrust–time traces (P~t and F~t), respectively. t₁ and t₂ are the initial moment and final moment of propellant burning, respectively. The total impulse (I₀) of the propellant can be obtained according to Equation (1):

$$I_0={\displaystyle \underset{t_1}{\overset{t_2}{\int }}F(t)dt}$$

(1)

After that, the average specific impulse ($\overline{I_s}$) and the average pressure of the propellant during the whole burning process of the propellant ($\overline{P}$) are obtained using Equation (2) and Equation (3), respectively.

$$\overline{I_s}=I_0/M_p$$

(2)

$$\overline{P}={\displaystyle \underset{t_1}{\overset{t_2}{\int }}P(t)}dt /(t_2-t_1)$$

(3)

The burning time of the propellant is divided into n (n > 1) segments, and each segment is denoted as s_i ($i=1,2,3,\dots ,n$). The duration of s_i is denoted as $\Delta$t_i, m_i is the mass of the propellant burned during $\Delta$t_i, and $\overline{P_i}$ is the average pressure of s_i. There is a relationship, as shown in Equation (4):

$$I_{0i}=m_i·I_s(\overline{P_i})$$

(4)

where I_0i is the total impulse of s_i, $I_s(\overline{P_i})$ is the specific impulse at the corresponding $\overline{P_i}$, and $I_s\left(\overline{P_i}\right)$ is a function of $\overline{P_i}$ and can be obtained through the approach given later. Additionally, I_0i and $\overline{P_i}$ can be obtained using Equations (5) and (6):

$$I_{0i}={\displaystyle \underset{\Delta t_i}{\int }F(t)dt}$$

(5)

$$\overline{P_i}={\displaystyle \underset{\Delta t_i}{\int }Pdt}/\Delta t_i$$

(6)

Then, m_i is obtained using Equation (7):

$$m_i=I_{0i}/I_s(\overline{P_i})$$

(7)

e_n is the web thickness of the propellant that was burned off during $\Delta$t_n. According to the geometric parameters of the propellant grain, e_n can be calculated using Equation (8):

$$m_n=\pi \left(D^2-{\left(D-2e_n\right)}^2\right)L{\rho }_p/4$$

(8)

Correspondingly, e_n−1, e_n−2, …, e₁ can be solved sequentially through the following relationships:

$$m_{n-1}=\pi \left({\left(D-2e_n\right)}^2-{(D-2e_n-2e_{n-1})}^2\right)L{\rho }_p/4$$

(9)

$$m_{n-2}=\frac{\pi }{4}\left({(D-2e_n-2e_{n-1})}^2-{(D-2e_n-2e_{n-1}-2e_{n-2})}^2\right)L{\rho }_p$$

(10)

${\rho }_p$ in Equations (8)–(10) is the density of the propellant grain.

In addition, the pressure at each moment of $\Delta$t_i is considered to be equal to $\overline{P_i}$, and the burning rate (r_i) corresponding to $\overline{P_i}$ is

$$r_i=e_i/\Delta t_i$$

(11)

2.2.2. Actual Specific Impulse

In Equation (4), the actual specific impulse (I_s(actual)) at the corresponding pressure is unknown, but it is essential for subsequent data processing. Assuming that the actual specific impulse follows a consistent trend with the theoretical specific impulse (I_{s(theoretical)}) as a function of pressure, the following method is proposed to determine the specific impulse at the corresponding pressure:

(1)

Use the chemical thermodynamic calculations based on the tested propellant’s composition to obtain the theoretical specific impulse values at a series of different pressures.

(2)

Fit the pressures and corresponding theoretical specific impulse values using an appropriate mathematical function, which will result in a function depicting the variation of the theoretical specific impulse with the pressure (I_{s(theoretical)}$~$P), as shown by curve 1 in Figure 4.

(3)

Use Equations (2) and (3) to calculate the average specific impulse and the average pressure values, respectively. Locate the corresponding point ($\overline{P},\overline{I_s}{}_{(actual)}$) in Figure 4.

(4)

Since the actual specific impulse should be lower than the theoretical specific impulse, shift curve 1 in Figure 4 downward in parallel until it passes through the point ($\overline{P},\overline{I_s}{}_{(actual)}$). This adjustment will produce curve 2, representing the relationship between the actual specific impulse and the pressure (I_s(actual)$~$P).

3. Experimental Results and Analysis

3.1. Experimental Samples

The experimental samples were GAP-based composite propellants and SQ2 double-base propellants. The main components of the SQ2 double-base propellants were nitrocellulose, nitroglycerin, dinitrotoluene, calcium carbide, and lead oxide. It is important to note that the formulation of the SQ2 propellants used in the experiments was consistent, but the grain dimensions varied. The binder systems of the GAP-based composite propellants were based on GAP, with other key components including aluminum powder and oxidizer AP/HMX. Similarly, the formulations of the GAP-based composite propellants remained uniform, but the grain dimensions varied in the experiments.

Table 1. Performance and parameters of test propellant grains.

Types of Propellants	Densities (g·cm3)	Experiment No.	d (mm)	D (mm)	L (mm)	dt1 (mm)	dt2 (mm)
GAP composite propellant	1.735	GAP-1	20.0	54.0	170.0	7.80	8.10
		GAP-2	20.0	54.0	184.0	8.40	8.90
SQ2 double-base propellant	1.610	SQ2-1	35.0	73.3	180.0	8.00	9.42
		SQ2-2	30.0	63.0	200.0	8.00	8.55

provides an overview of the geometrical dimensions and densities of the propellant grains, as well as the nozzle throat diameters. The outer diameter, inner diameter, and length of the propellant grain are represented by D, d, and L, respectively. Moreover, the initial and final throat diameters of the nozzle are represented by d_t1 and d_t2, respectively.

3.2. Processing of Test Data for GAP-1

The P~t and F~t curves of GAP-1, GAP-2, SQ2-1 and SQ2-2 were obtained using the Impulse Method. Taking GAP-1 as an example, the data processing method of the Impulse Method described in Section 2.2.1 was used to calculate the burning rates at a range of pressures.

Figure 5a shows the P~t and F~t curves of GAP-1, where the pressure and thrust of the working section both increased with time. The curves between the initial and final moments of propellant burning in Figure 5a were extracted, resulting in the curves shown in Figure 5b, which were the bases for the subsequent data processing to obtain the burning rates.

According to the propellant formula, the theoretical specific impulses of the GAP-based propellant at different pressures were obtained via a chemical thermodynamic calculation, and they are shown in

Table 2. Theoretical specific impulses of the SQ-2 double-base propellant under different pressures.

Pressure (MPa)	Theoretical Impulse (m/s)
8	2582
10	2617
15	2666
20	2691
25	2706
30	2716
40	2728
50	2737
60	2742
70	2746
80	2749

. The data in Table 2 were fitted, and the result is shown in Figure 4. The fitting function was the following (curve 1 in Figure 4):

$$I_s=2709\times e^{1.9\times {10}^{-4}P}-365.6\times e^{-0.13P}$$

(12)

Based on Figure 5b, the average pressure of the whole working process of the motor was calculated using Equation (3), which was $\overline{P}$ = 17.92 MPa. The total impulse was calculated using Equation (1), which was $I_0$ = 1057.88 N·s. The average specific impulse of the propellant was 1978.04 m/s. It should be noted that the actual specific impulse of the propellant is normally lower than the theoretical specific impulse. In this test, the expansion efficiency of the nozzle was not considered, so the actual specific impulse was much lower. The fitting function was modified according to the calculated actual average pressure and the average specific impulse values, and the corrected result is shown as follows (curve 2 in Figure 4):

$$I_s=2709\times e^{1.9\times {10}^{-4}P}-365.6\times e^{-0.13P}-580.73$$

(13)

Then, the burning period was divided into 30 segments, each of which was 27.8 ms. According to data in Figure 5b and the processing method in Section 2.2.1, the following parameters were calculated in order: the total impulse within each time period (I_0i), the average pressure during each time period ($\overline{P_i}$), the mass burned in each time period (m_i), and the web thickness that was burned off at each time period (e_i). Notably, $Is(\overline{P_i})$ in Equation (7) is the actual average specific impulse corresponding to each average pressure. Thus, the burning rate r_i corresponding to each $\overline{P_i}$ could be calculated using Equation (11). The calculation results are shown in

Table 3. The burning rate of GAP-1 tested using the Impulse Method.

Pressure (MPa)	Burning Rate (mm/s)	Burned Mass (g)	Pressure (MPa)	Burning Rate (mm/s)	Burned Mass (g)
5.82	11.70	5.88	16.00	20.73	17.57
6.12	12.06	6.27	17.18	21.39	18.78
6.53	12.58	6.77	18.29	21.90	19.91
6.99	13.16	7.32	19.48	22.50	21.16
7.46	13.81	7.96	20.84	23.23	22.61
7.98	14.47	8.63	22.22	23.86	24.03
8.54	15.10	9.33	23.64	24.55	25.58
9.15	15.73	10.07	25.18	25.35	27.31
9.76	16.34	10.84	26.82	26.13	29.13
10.44	16.97	11.66	28.55	26.99	31.11
11.20	17.68	12.59	30.45	28.01	33.39
12.01	18.33	13.53	32.58	29.14	35.93
12.93	18.97	14.51	34.99	30.45	38.84
13.89	19.61	15.49	37.55	32.01	42.26
14.91	20.19	16.53	40.36	33.42	45.68

.

Since the specific impulses used in the calculation process were corrected values (actual specific impulse), if the corrected values were reasonable, the sum of the calculated masses of the burned-off propellant for each segment should be equal to M_p. According to the burning masses in Table 3, the sum was 590.68 g. Based on the dimensions of the propellant grain, the real mass was 591.16 g. Obviously, the two values were almost equal. This showed that the calculation method and process were reasonable.

3.3. Burning Rate Test Results and Analysis

Figure 6 displays the P~t curves and F~t curves of GAP-2. The burning rates at various pressures of all the samples in experiment, obtained through meticulous data processing with the Impulse Method, are partially listed in

Table 4. Burning rates of the propellant grains.

GAP-1		GAP-2		SQ2-1		SQ2-2
Pressure /MPa	Burning Rate/mm/s	Pressure /MPa	Burning Rate/mm/s	Pressure /MPa	Burning Rate/mm/s	Pressure /MPa	Burning Rate/mm/s
5.87	11.71	5.31	12.40	8.87	11.16	9.00	11.15
6.62	12.49	6.05	12.59	9.64	11.35	9.68	11.28
7.55	13.49	6.87	13.09	10.48	11.63	10.42	11.49
8.64	14.67	7.78	13.86	11.37	11.90	11.23	11.73
9.85	15.96	8.83	14.80	12.34	12.18	12.08	12.00
11.30	17.31	10.05	15.86	13.39	12.52	13.00	12.27
13.02	18.66	11.45	16.98	14.15	12.70	13.95	12.58
13.98	19.34	12.23	17.56	15.78	13.27	15.59	13.09
16.08	20.67	13.95	18.73	16.94	13.81	16.25	13.32
18.36	21.98	15.89	19.90	18.00	14.25	17.92	13.96
20.00	22.70	18.09	21.08	19.55	14.91	20.81	15.48
22.27	23.87	20.00	22.20	21.65	15.95	21.69	16.08
25.21	25.06	21.77	22.91	22.71	16.59	22.41	16.63
26.85	25.68	23.10	23.57	23.54	17.19	23.09	17.07
28.57	26.77	24.44	24.18	24.07	17.59	24.01	17.50
30.45	27.58	25.83	24.83	24.32	17.81	24.50	17.98
32.56	28.63	27.40	25.60	24.49	17.99	25.19	18.10
34.96	29.94	29.14	26.46	24.50	18.00	26.45	18.85
40.28	32.72	30.97	27.32	24.41	18.04	27.59	19.58

.

Figure 6a displays the P~t and F~t curves of GAP-2. It can be observed that the thrust and pressure gradually increased over time during propellant burning, reaching their peak values when the propellant grains were completely burned out. The maximum pressure achieved was 33.65 MPa. Following this peak, the pressure and thrust experienced a rapid decline. Furthermore, the burning time was measured to be 0.8968 s.

The comparison of the burning rates between two grains, GAP-1 and GAP-2, is illustrated in Figure 6b. It can be observed that both groups of burning rates increased with the increasing pressure, and at the same pressure, the burning rates were nearly equal. The maximum difference in the burning rate at the same pressure was less than 1 mm/s, as indicated by the data in Table 2, demonstrating a high level of reproducibility. Additionally, the burning rate of the GAP composite propellant at 20 MPa, obtained using the Standard Motor Method and the Impulse Method, was consistent. The burning rate obtained using the Standard Motor Method was 22.6 mm/s, and the Impulse Method yielded values of 22.2 mm/s and 22.7 mm/s, respectively. This consistency indicated that the two types of methods had comparable accuracy for testing the burning rate of a solid propellant.

The P~t and F~t curves of the SQ2-1 and SQ2-2 are shown in Figure 7a and Figure 7b, respectively. Similar to the trends of GAP-1 and GAP-2, it can be observed in Figure 7b that the thrust and pressure increased with time during propellant burning. The maximum pressure achieved by SQ2-2 was 28.06 MPa when the grain burned out, and the burning time was 1.2690 s.

In Figure 7a, the burning time of SQ2-1 was measured to be 1.3638 s. Initially, both the pressure and thrust increased with time. However, in the final period, the pressure tended to stabilize, while the corresponding thrust continued to incrementally increase. Notably, erosion of the nozzle was observed, indicated by an increase in the throat diameter from 8.10 mm before the test to 9.42 mm after the test, explaining why the pressure reached a plateau. As shown in Equation (14), the increasing A_t resulted in incremental thrust under constant pressure.

$$F=C_{F}P{A}_t$$

(14)

where A_t is the cross-sectional area of the nozzle throat; C_F is the thrust coefficient; and P is the chamber pressure.

Figure 8 shows the comparison of the burning rates obtained from testing SQ2-1 and SQ2-2. It can be seen that the burning rate increased with the increasing pressure for both samples. Furthermore, at the same pressure, the burning rates of SQ2-1 and SQ2-2 were found to be nearly equal. By comparing the burning rates of SQ2-1 and SQ2-2 at the same pressure presented in Table 4, it was determined that the maximum deviation was less than 0.3 mm/s, highlighting the high level of reproducibility.

According to the data for SQ2-1 in Table 4, the burning rates at similar pressures (24.50 MPa) during different periods were determined to be 17.99 mm/s, 18.00 mm/s, and 18.04 mm/s. These results highlight the high reproducibility of the burning rates at nearly equal pressures. Similarly, the burning rate of SQ2-2 at 24.50 MPa was found to be 17.98 mm/s, as shown in the data for SQ2-2. It can be observed that the burning rates, whether they were obtained from a single test or multiple tests, were almost equal at the same pressure. Therefore, it is concluded that the Impulse Method is both feasible and reliable. Additionally, the erosion of the nozzle only affects the trend of the pressure change during testing with the Impulse Method, without significantly impacting the burning rate of the propellant.

3.4. Rationality of the Method for Determining I_s(actual)

In addition to the Impulse Method, the Mass Flow Rate Method [28,29] is widely employed for testing the burning rate of a solid propellant in solid rocket motors. This method involves generating an incremental P~t curve with the burned propellant grain (as depicted in Figure 2), in which the inner surface of the grain serves as the initial burning surface. Based on the principle that the gas mass flow rate at the nozzle throat is equal to the mass of the propellant that was burned off, and assuming that both the characteristic velocity and throat diameter are constant throughout the test process, the average pressure and the corresponding mass of propellant consumed can be determined using Equation (6) and Equation (15), respectively.

$$m_i=A_t{\displaystyle \underset{\Delta t_i}{\int }P(t)}dt/C^{*}=M_P{\displaystyle \underset{\Delta t_i}{\int }P(t)}dt/{\displaystyle \underset{t_1}{\overset{t_2}{\int }}P(t)}dt$$

(15)

Subsequently, the burning rates at different pressures can be derived according to the geometric parameters of the propellant grain using Equations (8)–(11). Figure 9 illustrates the underlying principles and steps involved in this method, while more detailed information regarding the data processing procedures can be found in relevant studies.

Similar to the Impulse Method, the Mass Flow Rate Method provides an alternative approach to determine the burning rate of a solid propellant in a compact thick-walled solid rocket test motor equipped with sensors. By analyzing the incremental P~t curve generated during the burning process, valuable insights into the propellant’s burning rate characteristics can be gained. The Mass Flow Rate Method relies on the assumption of a constant characteristic velocity and throat diameter, allowing for researchers to accurately calculate the mass of the propellant that was burned off. These then enable the derivation of burning rates at various pressures. Most remarkably, the P~t trace generated from one propellant sample can be used as test data for both the Impulse Method and the Mass Flow Rate Method to obtain the burning rates.

According to the P~t curve of GAP-1 in Figure 5a and the dimensional parameters in Table 1, the burning rates obtained using the Impulse Method and the Mass Flow Rate Method can be calculated, and they are illustrated in Figure 10a. It can be observed that the burning rates obtained using the Impulse Method were slightly greater than those obtained using the Mass Flow Rate Method at stages of lower pressures. At stages of higher pressures, the burning rates obtained using the Mass Flow Rate Method were greater than those obtained using the Impulse Method.

During the operation of the test motor, the characteristic velocity (C*) of the propellant slightly increased with the increasing pressure. Consequently, during the low-pressure stage, the actual C* of the propellant was slightly smaller than the average characteristic velocity ($\overline{C}*$), resulting in slightly higher actual burning rates compared to those obtained using the Mass Flow Rate Method. Conversely, during the high-pressure stage, the actual C* of the propellant was slightly larger than $\overline{C}*$, leading to slightly lower actual burning rates compared to those obtained using the Mass Flow Rate Method. Furthermore, minor nozzle erosion occurring in the later stages of testing caused a slight increase in the throat diameter. This increase had the small effect of reducing the burning rate determined using the Mass Flow Rate Method, bringing it closer to the actual burning rate. By analyzing the differences between the burning rates obtained using the two methods, as depicted in Figure 10, it can be observed that the Impulse Method’s burning rates were consistent with the actual burning rates, which proved the rationality of the method for determining the actual specific impulse.

According to the P~t and F~t curves of SQ2-1 shown in Figure 7a and the dimensional parameters provided in Table 1, the burning rates were calculated using the two methods and are presented in Figure 10b. During the low-pressure stage, it was observed that the burning rates obtained using the Mass Flow Rate Method were greater than those obtained using the Impulse Method. However, during the high-pressure stage, the burning rate obtained from the Mass Flow Rate Method was smaller compared to that obtained from the Impulse Method.

For the SQ2-1 grain, theoretically, the total web thickness (e_a) and burning time (t_a) were constant, at 19.15 mm and 1.3638 s, respectively, regardless of whether the Impulse Method or the Mass Flow Rate Method was used to determine the burning rate. This means that the average burning rate (r_avg) throughout the combustion process was constant, which was $r_{avg}=e_a/t_a\approx$ 14.04 mm/s. However, due to severe nozzle erosion, the burning rates of SQ2-2 obtained using the Mass Flow Rate Method decreased significantly with the increasing pressure at a high pressure, which resulted in lower burning rates from the Mass Flow Rate Method than those obtained from the Impulse Method. Since the average burning rate of SQ2-2 obtained using the two methods was the same, which was 14.04 mm/s, the burning rates at a low pressure that were obtained using the Mass Flow Rate Method would inevitably be larger than those obtained using the Impulse Method.

In addition, the characteristic velocity of both the SQ2 double-base propellant and the GAP-based composite propellant tended to increase with the pressure, but they were affected by pressure in different degrees, which had a different effect on the pattern of difference between subfigures a and b in Figure 10. However, this different effect was considered to be smaller compared to the effect of severe nozzle erosion. In summary, the severe nozzle erosion of SQ2-2 was considered to be the main cause of the results in Figure 10b, while the variation in the characteristic velocity with pressure was considered to be the main cause of the results in Figure 10a.

Notably, a phenomenon was observed in which the Mass Flow Rate Method’s burning rates of different periods were different at the same pressure, which contravened the fundamental principles of propellant combustion. The data for d_t1 and d_t2 in Table 1 show that the nozzle erosion of SQ2-1 was severe, so it was believed that the phenomenon in the burning rate was attributed to the neglect of nozzle erosion in the Mass Flow Rate Method. However, the Impulse Method’s burning rates for different periods were nearly equal at the same pressures. These findings indicate that the burning rate obtained using the Impulse Method is not influenced by nozzle erosion.

4. Conclusions

(1)

Two types of propellants were subjected to burning rate tests using the Impulse Method. The burning rates were nearly equal for propellants of the same type at the corresponding pressures, demonstrating the feasibility of the method.

(2)

The Impulse Method is efficient for testing the burning rate of a solid propellant. It allows one to obtain the burning rate of a solid propellant at any pressure within an increased pressure range through a single test. The burning rates obtained using the Impulse Method and the Standard Motor Method at the same pressure were consistent, indicating the comparability between the Impulse Method and the Standard Motor Method. Furthermore, the Impulse Method eliminates the need to consider errors introduced by nozzle erosion.

(3)

A comparative analysis was conducted on the burning rates obtained through the Mass Flow Rate Method and the Impulse Method. The results provide evidence supporting the rationality of the approach in determining the actual specific impulse.