Design method of tubular grain for impulse method burning rate test

Abstract

The invention provides a method for determining the size of a tubular grain for testing the burning rate through the impulse method, and the method comprises the following steps: establishing a relationship between the pressurization rate and the burning characteristic of a propellant and a mathematical relationship between the pressurization rate and a tubular charging size parameter, and revealing the influence of the inner hole burning tubular charging size on the pressure intensity change. Different requirements of impulse method combustion rate tests of propellants with different combustion characteristics on the sizes of the explosive types are determined. And in combination with constraints such as pressurization rate limitation, pressure intensity test range and erosion-resistant combustion, a tubular charging design criterion considering both test precision and safety is provided.

Description

Technical Field

[0001] This invention belongs to the field of dynamic testing technology for the combustion performance of solid propellants; it mainly relates to a design method for tubular propellant grains used in impulse burning rate testing. Background Technology

[0002] The burning rate of solid propellants is a key parameter characterizing their combustion properties and directly affects the performance of rocket engines. The burning rate is defined as the distance the propellant charge's burning surface recoils per unit time along its normal direction. It determines the energy release rate and serves as the basis for calculating combustion performance parameters such as the burning rate coefficient, pressure index, temperature sensitivity coefficient, and erosion ratio.

[0003] The literature "Impulse Method for Testing Dynamic Burning Rate and Pressure Index of Solid Propellants under High Pressure" (Journal of Explosives and Pyrotechnics, 2019(03), 278-283) and the patent "Validity Judgment of Original Data for Burning Rate Test of Solid Propellants using Impulse Method" ZL202110283176.2, 2022.3 propose an efficient test method for dynamic testing of burning rate of solid propellants using impulse method. This method employs a tubular charge with internal combustion to obtain thrust-time and pressure-time curves that increase over time. A single test can measure the burning rate at any pressure within the pressurization range. The premise of impulse method burning rate testing is that the exposed internal hole can ignite synchronously, and the burning rate can respond promptly to changes in pressure. ZL202110283176.2, 2022.3 provides the criteria for ignition synchronization. The tubular charge with internal combustion can ensure the ignition synchronization of the exposed initial combustion surface, and also has advantages such as simple structure and efficient burning rate coupling calculation. However, no specific method for determining the size of the tubular propellant grain is given, resulting in a small pressurization range for propellants with low burning rate and low pressure index, which is insufficient to achieve the purpose of efficient burning rate testing. On the other hand, although high burning rate or high pressure index propellants have a large pressurization range, there is a problem that the propellant combustion cannot respond to changes in pressure in a timely manner. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies and to address the issues that low-burning-rate and low-pressure-index propellants have a small pressurization range, which is insufficient for efficient burning rate testing, and high-burning-rate or high-pressure-index propellants have a large pressurization range, but the propellant combustion cannot respond promptly to pressure changes, this invention provides a design method for tubular propellant grains used in impulse-based burning rate testing.

[0005] A method for designing tubular propellant charges for impulse-based burning rate testing includes the following steps: Step 1: Set the nozzle throat diameter Initial value; According to nozzle throat diameter Calculate the propellant burning surface area ; Step 2: Calculate the minimum propellant burning surface area based on requirements. and obtain the lowest pressure. Set the initial value of the inner diameter d of the propellant column; Step 3: Calculate the propellant grain length based on the minimum propellant burning surface area. ; Minimum propellant burning surface area satisfy: ; according to Calculate the length of the propellant column ; If the length of the propellant column ≥200mm is considered If the diameter is too large, the inner diameter of the tubular charge will be too small; in this case, the nozzle throat diameter dt will be reduced in increments of 0.2 mm, and the process will return to step 1 and be recalculated. If the length of the propellant column If the diameter is less than 200 mm, then the inner diameter d of the propellant column and the nozzle throat diameter are obtained. and the length of the medicine column ; Step 4: Determine the outer diameter D of the propellant column based on the range of the pressure index; Step 5: Based on the obtained outer diameter D, inner diameter d, and length of the propellant column... Calculate the pressurization rate of the tubular charge; Differentiating the transient equilibrium pressure formula in the combustion chamber with respect to time, we obtain the boost rate. They are represented as follows: (1) (2) In the formula, The rate of change of the burning surface, d is the inner diameter of the propellant column. Let be the thickness of the meat burned at time t, 0 ≤ ≤(Dd) / 2, where D is the outer diameter of the propellant column and L is the length of the propellant column; i.e., boost rate for: (3) When the boost rate is too high, the combustion rate may not respond promptly to changes in pressure. Therefore, the boost rate must be limited. A critical value for the boost rate can be used for limitation, resulting in: (4) in, Currently, the flesh is thick; (5) when Greater than or equal to If the pressure rate is 80%, it is considered that the pressurization rate is too high. Therefore, the inner diameter d of the propellant column is reduced by a step size of 2 mm, and the process jumps to step 3 to recalculate. when Less than When the pressure is 80% of the required level, it is considered that the pressurization rate can meet the requirement of timely pressure response. At this point, the inner diameter of the propellant grain (d) and the nozzle throat diameter... The length L and outer diameter D of the propellant column are the dimensions of the tubular propellant charge, which can meet the requirements of timely pressure response.

[0006] Furthermore, the step of obtaining the critical value of the boost rate is as follows: Step 5-1: Obtain the rate of change of propellant burning rate under the presence of pressurization rate; The rate of change of burning rate is the ratio of the difference between static and dynamic pressure burning rates to the response time of the dynamic pressure burning rate; the static pressure burning rate of the propellant under constant pressure is... The dynamic pressure burning rate corresponding to a rapid change in pressure is: Dynamic pressure combustion rate from Become The process takes time ,Right now The response time of dynamic pressure combustion rate; This is the combustion rate coefficient; Pressure; The rate of change of combustion speed for: ; in n is the burn rate coefficient; n is the pressure exponent; It characterizes the degree of hysteresis in the response of dynamic pressure burning rate to pressure changes; Step S5-2: Rearrange the expression for the rate of change of burning rate to obtain the dynamic pressure burning rate and the dynamic relationship between the rate of change of burning rate and pressure. The rate of change of combustion rate By rearranging the terms, we obtain the dynamic relationship between dynamic pressure-induced combustion rate and the rate of change of combustion rate with pressure:

[0007] The dynamic relationship between dynamic pressure-induced combustion rate and the rate of change of combustion rate with pressure is a classical first-order dynamic differential equation, while the dynamic relationship between dynamic pressure-induced combustion rate and the rate of change of combustion rate with pressure is a first-order inertial element; when the pressure is constant, i.e. At that time, the Vieri formula is equivalent to the dynamic relationship between dynamic pressure burning rate and the rate of change of burning rate and pressure. Step S5-3: Determine the boundary conditions for dynamic pressure burning rate and timely response to pressure changes; When the boost rate When it is large, because Since it is greater than zero, the dynamic relationship between dynamic pressure burning rate and the rate of change of burning rate and pressure is derived from the dynamic formula. Therefore, the dynamic pressure combustion rate at the boost rate is lower than the static pressure combustion rate at steady state, so it manifests as a decrease in the dynamic pressure combustion rate at the boost rate. When the pressure is high, the dynamic pressure combustion rate response will lag. To ensure that the dynamic pressure burning rate responds promptly to changes in pressure, the boundary conditions for the dynamic pressure burning rate to respond promptly to changes in pressure are determined as follows:

[0008] When the boundary condition requirement of dynamic pressure burning rate responding to pressure change in a timely manner is met, the dynamic pressure burning rate can respond to pressure change in a timely manner. That is, the Vieri formula and the dynamic relationship between dynamic pressure burning rate and the rate of change of burning rate and pressure are equivalent. Step S5-4: Based on the boundary conditions of dynamic pressure combustion rate and timely response to pressure changes, differentiate the Vieri formula with respect to time to obtain the boundary conditions of the boost rate. Differentiating Vieri's formula with respect to time yields:

[0009] Substituting the result obtained by differentiating the Vieri formula with respect to time into the boundary conditions for dynamic pressure-burning rate and timely response pressure changes, the boundary conditions for the boost rate are derived as follows:

[0010] Step S5-5: Determine the critical value and range of boost rate; Take the product of the rate of change of burning rate and the dynamic pressure burning rate response time as less than the dynamic pressure burning rate. This is the critical condition, i.e. The critical value for obtaining the boost rate is calculated. ,Right now The pressurization rate range in which the propellant dynamic pressure burning rate responds promptly to pressure. Therefore, the critical value for boost rate is:

[0011] Since the minimum time interval for the impulse method combustion rate test is 0.05s, the required combustion rate response time is... The time limit must not exceed 0.005 s. This study adopted... The baseline value is 0.005s, which means that the propellant burning rate is considered to respond promptly to pressure changes within 0.005s. This value ensures the applicability of the critical pressurization rate calculation and meets the burning rate testing requirements of most propellants. The unit of critical pressurization rate is MPa / s.

[0012] Furthermore, the nozzle throat diameter The initial value is 7mm. If the throat diameter is too small, it is easy to cause nozzle blockage during the experiment; if the throat diameter is too large, it will be difficult to establish the initial pressure. Setting the nozzle throat diameter to 7mm can avoid both situations.

[0013] Furthermore, the goal of obtaining the lowest pressure The steps are as follows: Minimum ignition rate Burn rate coefficient Pressure index n, propellant density Minimum value of propellant burning surface area nozzle throat area and propellant characteristic velocity Substituting the transient equilibrium pressure expression in the combustion chamber, the minimum pressure is obtained. Minimum pressure Lower than the minimum required pressure .

[0014] Furthermore, the step of determining the outer diameter D of the propellant column based on the range of the pressure index is as follows: The first candidate outer diameter is calculated based on the transient equilibrium pressure formula. ; When the pressure index n ≤ 0.38, calculate the second candidate outer diameter. for: ,in This is the thickest part of the body; to ensure sufficient burning time, the thickest part of the body is at this point. ≥15mm; ≥15mm; If the thickness of the first tissue is 15mm, then the outer diameter D of the drug cartridge is: ; When the pressure index is in the range of 0.38 < n ≤ 0.55, calculate the second candidate outer diameter. for: , This is the second thickest part of the meat; to ensure sufficient burning time, the range of the second thickest part is as follows: ≥20mm; If the second flesh thickness is 20mm, then the outer diameter D of the drug cartridge is: ; If the pressure index n>0.55, the tubular charge cannot meet the design requirements.

[0015] The dimensions of the tubular charge designed according to a method for determining the dimensions of tubular propellant charges for impulse burning rate testing are as follows: When the pressure index is less than or equal to 0.38: the propellant length L is greater than or equal to 150 mm, the nozzle throat diameter is less than or equal to 7.0 mm, and the flesh thickness ranges from [15 mm to 20 mm]. When the pressure index ranges from (0.38 to 0.55): the propellant length L is less than 150 mm, the nozzle throat diameter is greater than 7.0 mm, and the flesh thickness is greater than or equal to 20 mm.

[0016] The beneficial effects of this invention are as follows: This invention achieves quantitative design of the pressure range and pressurization rate of tubular propellants, establishes impulse-based tubular propellant design criteria, and effectively avoids the problem of traditional propellant design processes failing to simultaneously consider pressure range and pressurization rate. Simultaneously, the established quantitative calculation model for propellant dimensions provides targeted propellant design schemes for propellants with low to medium pressure indices and medium to high pressure indices, and provides recommended dimensions for both types of propellants. Attached Figure Description

[0017] Figure 1 The Pt curve for medium- and low-burning-rate propellants; Figure 2 A graph showing the pressurization rate of propellants with medium and low burning rates; Figure 3 This is a schematic diagram of a propellant grain for medium and low burning rates. Figure 4 Pt curves for medium-to-high burning rate propellants; Figure 5 A graph showing the pressurization rate of medium-to-high burning rate propellants; Figure 6 This is a schematic diagram of a medium-to-high burning rate propellant grain. Detailed Implementation The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0018] This is the lowest pressure. For maximum pressure, is the burning rate coefficient; n is the pressure exponent; C* is the propellant characteristic velocity, which is the given requirement data. Transient equilibrium pressure in the combustion chamber for: (1) Step 1: Determine the nozzle throat diameter ; Nozzle throat diameter It is 7 mm; typically, the throat diameter of the nozzle is designed for the impulse method tubular charge. Both are around 7mm. If the throat diameter is too small, it can easily lead to nozzle blockage during the experiment; if the throat diameter is too large, it will be difficult to establish the initial pressure. Setting the nozzle throat diameter to 7mm can avoid both situations. Therefore, the initial nozzle throat diameter is set to 7mm. It is 7mm; According to nozzle throat diameter Calculate the propellant burning surface area ; Step 2: Based on the minimum pressure specified in the requirements. Burn rate coefficient Pressure index n and propellant characteristic velocity The minimum propellant burning surface area is calculated based on the transient equilibrium pressure expression in the combustion chamber. Set the initial value of the inner diameter d of the propellant column; the initial value of the inner diameter d of the propellant column is 20mm; Minimum ignition rate Burn rate coefficient Pressure index n, propellant density Minimum propellant burning surface area nozzle throat area and propellant characteristic velocity Substituting the transient equilibrium pressure expression in the combustion chamber, the minimum pressure is obtained. ; To meet the requirements, the minimum propellant burning surface area The corresponding pressure is lower than the minimum required pressure. ; Step 3: Calculate the propellant charge length based on the erosion and combustion inhibition criterion. ; According to the erosion-burning suppression criterion, the initial burning surface satisfies... Thus, the length of the propellant column can be calculated. ; If the length of the propellant column ≥200mm is considered If the diameter is too large, the inner diameter of the tubular propellant column will be too small; in this case, the nozzle throat diameter dt will be reduced in increments of 0.2 mm, and the process will return to step 1. If the length of the propellant column Less than 200mm, to obtain a tubular charge propellant column d and nozzle throat diameter and the length of the medicine column Then proceed to step 4; Step 4: Calculate the outer diameter D of the propellant column; Calculate the first candidate outer diameter according to Formula 1. ; When the pressure index n ≤ 0.38, and the wall thickness e ≥ 15mm while ensuring combustion time, the second candidate outer diameter... for: , where e represents flesh thickness; The outer diameter D of the propellant column is: ; When the pressure index is 0.38 < n ≤ 0.55, and the wall thickness e ≥ 20 mm while ensuring combustion time, then the second candidate outer diameter... for: e represents thick flesh; The outer diameter D of the propellant column is: ; If the pressure index n>0.55, the tubular charge cannot meet the design requirements; Step 5: Calculate and obtain the outer diameter D, inner diameter d, and length of the propellant column. Calculate the pressurization rate of the tubular charge; Differentiating equation (1) with respect to time, the boost rate dP / dt is expressed as equation (2) and equation (3), respectively.

[0019] (2) (3) In the formula, The rate of change of the burning surface, d is the inner diameter of the propellant column. Let be the thickness of the meat burned at time t, 0 ≤ ≤(Dd) / 2, where D is the outer diameter of the propellant column and L is the length of the propellant column; i.e., boost rate for:

[0020] When the boost rate is too high, the combustion rate may not respond promptly to changes in pressure. Therefore, the boost rate must be limited. A critical value for the boost rate can be used for limitation, resulting in:

[0021] in, Currently, the flesh is thick; (5) when Greater than or equal to If the pressure rate is 80%, it is considered that the boost rate is too high. Therefore, the pressure is reduced by d in steps of 2mm, and the process jumps to step 3 to recalculate. when Less than When the pressure is 80%, it is considered that the pressurization rate can meet the requirement of timely pressure response. At this time, the inner diameter d, nozzle throat diameter dt, L, and D are the dimensions of the tubular charge, which can meet the requirement of timely pressure response. The specific steps for obtaining the critical value of the boost rate are as follows: For solid propellants, the relationship between the burning rate and pressure under static pressure is expressed by the Vieri formula:

[0022] In the formula, This refers to the burning rate under static pressure. This is the combustion rate coefficient; Pressure; The pressure index; the Vieri formula reveals the index When the pressure changes, the burning rate also changes accordingly; pressure index At that time, the burning rate does not change with pressure. In the solid propellant impulse burning rate test, a tubular propellant grain with internal combustion is used to obtain pressure and thrust curves that increase over time, and the burning rate under pressure within the pressurization range is tested. If the pressurization rate is greater than the critical value, the propellant combustion cannot respond to the pressure change in time, that is, the burning rate cannot respond to the pressure change in real time. At this time, the change in burning rate will lag behind the change in pressure. Step 5-1: Obtain the rate of change of propellant burning rate under the presence of pressurization rate; Static pressure burning rate is the burning rate of propellant under constant pressure, and it is only related to the magnitude of the constant pressure. Dynamic pressure burning rate is the burning rate corresponding to a rapidly changing pressure, and it depends not only on the current pressure but also on the rate of pressure change. The static pressure burning rate of the propellant under constant pressure is: ; The dynamic pressure burning rate corresponding to rapid pressure change is: Dynamic pressure combustion rate from Become The process takes time ,Right now The response time of dynamic pressure combustion rate; The rate of change of combustion speed for: ; in n is the burn rate coefficient; n is the pressure exponent; It characterizes the degree of hysteresis in the response of dynamic pressure burning rate to pressure changes; Step S5-2: Rearrange the expression for the rate of change of burning rate to obtain the dynamic pressure burning rate and the dynamic relationship between the rate of change of burning rate and pressure. The rate of change of combustion rate By rearranging the terms, we obtain the dynamic relationship between dynamic pressure-induced combustion rate and the rate of change of combustion rate with pressure:

[0023] The dynamic relationship between dynamic pressure-induced combustion rate and the rate of change of combustion rate with pressure is a classical first-order dynamic differential equation, while the dynamic relationship between dynamic pressure-induced combustion rate and the rate of change of combustion rate with pressure is a first-order inertial element; when the pressure is constant, i.e. At that time, the Vieri formula is equivalent to the dynamic relationship between dynamic pressure burning rate and the rate of change of burning rate and pressure. Step S5-3: Determine the boundary conditions for dynamic pressure burning rate and timely response to pressure changes; When the boost rate When it is large, because Since it is greater than zero, the dynamic relationship between dynamic pressure burning rate and the rate of change of burning rate and pressure is derived from the dynamic formula. Therefore, the dynamic pressure combustion rate at the boost rate is lower than the static pressure combustion rate at steady state, so it manifests as a decrease in the dynamic pressure combustion rate at the boost rate. When the pressure is high, the dynamic pressure combustion rate response will lag. To ensure that the dynamic pressure burning rate responds promptly to changes in pressure, the boundary conditions for the dynamic pressure burning rate to respond promptly to changes in pressure are determined as follows:

[0024] When the boundary condition requirement of dynamic pressure burning rate responding to pressure change in a timely manner is met, the dynamic pressure burning rate can respond to pressure change in a timely manner. That is, the Vieri formula and the dynamic relationship between dynamic pressure burning rate and the rate of change of burning rate and pressure are equivalent. Step S5-4: Based on the boundary condition of dynamic pressure combustion rate responding to pressure change in a timely manner, the dynamic pressure combustion rate and the dynamic relationship between combustion rate change rate and pressure are differentiated over time to obtain the boundary condition of pressurization rate. Differentiating Vieri's formula with respect to time yields:

[0025] Substituting the result obtained by differentiating the Vieri formula with respect to time into the boundary conditions for dynamic pressure-burning rate and timely response pressure changes, the boundary conditions for the boost rate are derived as follows:

[0026] Step S5-5: Determine the critical value and range of boost rate; Take the product of the rate of change of burning rate and the dynamic pressure burning rate response time as less than the dynamic pressure burning rate. This is the critical condition, i.e. The critical value for obtaining the boost rate is calculated. ,Right now The pressurization rate range for the propellant's dynamic pressure burning rate in response to pressure; the response time of the dynamic pressure burning rate. It is a key parameter characterizing the dynamic response of the burning rate of solid propellants; The smaller the value, the stronger the propellant burning rate's response to pressure changes, and the larger the critical value of the pressurization rate; burning rate response time It is a key parameter for measuring the dynamic response characteristics of propellants; The smaller the value, the stronger the propellant burning rate's response to pressure changes, and the greater the critical value for responding to higher pressurization rates. In automatic control theory, the steady-state error of the first-order dynamic differential equation This indicates that "high-precision control" has been achieved, meaning that... At that time, the dynamic pressure burning rate can respond promptly to the boundary conditions of pressure changes and meet the requirements. In signal processing theory, minor components in a signal can be ignored when their proportion is less than 1%. For dynamic pressure-based combustion rate and timely response to pressure changes, the boundary condition is taken as... At that time, secondary components The influence can be ignored; in the theory of electronic information technology, nonlinear components can be approximated linearly when the input disturbance is <1%. For dynamic pressure combustion rate and timely response to pressure changes, the boundary conditions are taken as follows: At this time, the disturbance can be ignored. The overall impact; in engineering and science, a critical value of 1% is widely considered an extremely small threshold, and its rationale has been explained from a multidisciplinary perspective; therefore, the critical value for boost rate is chosen as... ,Right now The pressurization rate range in which the propellant burning rate responds promptly to pressure. Therefore, the critical value for boost rate is:

[0027] Since the minimum time interval for the impulse method combustion rate test is 0.05s, the required combustion rate response time is... The time limit must not exceed 0.005 s. This study adopted... The baseline value is 0.005s, which means that the propellant burning rate is considered to respond promptly to pressure changes within 0.005s. This value ensures the applicability of the critical pressurization rate calculation and meets the burning rate testing requirements of most propellants. The unit of critical pressurization rate is MPa / s. As can be seen from the critical value of the boost rate, the higher n is, the smaller the critical value of the boost rate. This is because the pressure exponent n reflects the sensitivity of the combustion rate r to the change in pressure P. The larger n is, the greater the fluctuation of the combustion rate with pressure, and the smaller the rate of pressure change that the combustion rate can respond to within a specified response time. That is, the critical value of the boost rate is negatively correlated with the pressure exponent n. This patent is described below with reference to specific embodiments: A certain low-to-medium burning rate propellant (1#) has a burning rate of 7.5 mm / s at 7 MPa, a pressure index n1=0.3, a density ρ1=1.8 g / cm3, and a characteristic velocity C*1=1500 m / s. The required pressure testing range is 7-20 MPa. A certain medium-to-high burning rate propellant (2#) has a burning rate of 14.3 mm / s at 7 MPa, a pressure index n1=0.51, a density ρ1=1.8 g / cm3, and a characteristic velocity C*1=1700 m / s. The required pressure testing range is 15-30 MPa.

[0028] The burning rate coefficients of the two propellants are a1 = 6.6 × 10⁻⁵ (m / s) / Pa 0.3 and a2 = 5.3 × 10⁻⁶ (m / s) / Pa 0.51.

[0029] Analysis based on equations (1) and (3) shows that the pressure and pressurization rate of tubular propellants during combustion are influenced by a combination of various propellant parameters. For a specific type of propellant, within a given pressure test range, there are often multiple feasible ranges for its propellant size configuration, rather than a single definite value. This characteristic requires us to comprehensively consider the interrelationships between various parameters during the design process: firstly, we need to determine the applicable boundary conditions for each size parameter (such as inner diameter, outer diameter, length, etc.) through theoretical calculations; then, we need to combine the actual burning rate characteristics of the propellant, structural strength requirements, and engineering constraints such as testing requirements to perform synergistic optimization of these parameters, and finally obtain reference values ​​for propellant size that meet the performance index requirements.

[0030] To standardize the design of tubular propellant charges, and based on experience in designing tubular propellant charges, the design process for tubular propellant charges is set as follows: (1) Since the throat diameter dt of most tubular charge designs using impulse method is around 7 mm, the initial throat diameter dt is set to 7 mm.

[0031] (2) The initial burning surface Abmin is calculated by equation (1) based on the lowest pressure Pmin in the pressure test range. During the experiment, the pressure range must cover the required pressure test range, so the pressure corresponding to the initial burning surface Abmin should be slightly lower than the required lowest pressure Pmin.

[0032] (3) Based on the erosion and combustion suppression criteria and design experience, the inner diameter d is set to 20 mm, and the initial burning surface meets the following requirements. Therefore, the propellant length L can be calculated based on the initial combustion surface. L must satisfy L < 200 mm. If L is too large, it means that d is too small. In this case, we should return to the first step to correct the nozzle throat diameter dt.

[0033] (4) To ensure a certain combustion time, for medium-low burning rate propellants n≤0.38, the wall thickness e≥15mm; for medium-high burning rate propellants 0.38<n≤0.55, the wall thickness e≥20mm. The larger value between the outer diameter calculated from the required maximum pressure Pmax and the outer diameter calculated from the wall thickness shall be taken as the outer diameter D.

[0034] (5) Calculate the pressurization rate according to the designed drug type using formula (4). When the pressurization rate is too high, increase the inner diameter d and shorten the length L until the pressurization rate and pressure range meet the requirements.

[0035] For the characteristics of the low-to-medium burning rate propellant #1, the tubular charge design was optimized. First, the nozzle throat diameter dt was set to 7mm. The experimental requirement was a minimum pressure of less than 7MPa. Calculations showed the initial burning surface should be less than 11806mm², and the propellant length L less than 188mm. Since the propellant length L was close to 200mm at this point, to avoid erosive combustion, the nozzle throat diameter was modified, reducing it to 6.8mm. With this reduced initial burning surface, the propellant length L was less than 11122mm², and the propellant length L less than 177mm. Therefore, the propellant length L was set to 170mm. Based on the maximum pressure being at least 20MPa, the outer diameter D was calculated to be at least 43.4mm. Considering the wall thickness e ≥ 15mm, the outer diameter D was set to 50mm. At this point, the length-to-diameter ratio was 8.5, which effectively suppressed erosive combustion. The design drawing is shown below. Figure 1 According to equation (3), the maximum pressurization rate of this size propellant column is calculated to be 18.0 MPa / s. According to equation (5), the critical pressurization rate at the highest pressure is calculated. It is 150 MPa / s, see Figure 2 During the combustion of propellants with medium and low pressure indices, the pressurization rate is much lower than the critical value, with a safety margin of 8.3 times. The pressure-time curves are shown below. Figure 3 At this point, the mass m of the medicine column is 504.4g, which meets the requirement of being less than 1kg.

[0036] To optimize the design of the tubular charge for the medium-to-high burning rate propellant #2, the nozzle throat diameter dt was initially set to 7 mm. The experimental requirement was a minimum pressure of less than 15 MPa, resulting in an initial burning surface of less than 7786.7 mm², a charge length L of less than 124 mm, and a charge length L = 120 mm. Given a minimum maximum pressure of 30 MPa, the maximum burning surface Abmax was calculated to be at least 10936 mm², meaning an outer diameter D of at least 29 mm. An outer diameter D = 30 mm was chosen, but this resulted in a wall thickness e of only 5 mm, resulting in a combustion time of only 0.15 s. Increasing the wall thickness beyond this would lead to a maximum pressure far exceeding the experimental requirements, resulting in an excessively large pressure variation range. Reducing this pressure variation range necessitates increasing the nozzle throat diameter dt. When the nozzle throat diameter dt is increased to 8.4 mm, the initial burning surface should be less than 11213 mm2. When the inner diameter d is still 20 mm, the propellant length L is less than 179 mm. The propellant length L is taken as 170 mm. To ensure a certain combustion time, the flesh thickness e is taken as 20 mm, that is, the outer diameter D is 60 mm. The final pressure reaches 127 MPa and the pressurization rate reaches 220 MPa / s, which greatly exceeds the required test range. At this time, further increasing the nozzle throat diameter will make it difficult to establish the ignition pressure. Therefore, it is chosen to increase the inner diameter d and shorten the length L. Therefore, the inner diameter d is increased in increments of 5 mm, and iterative calculation is performed according to steps (2)-(5). When d increases to 35 mm, D is 75 mm. The propellant length L is set to 80 mm. At this time, the pressure test range is 9.13 MPa-43.25 MPa, and the maximum pressurization rate is 78 MPa / s, which is much less than the critical pressure rate value. It is 169.6 MPa / s, see Figure 5 The safety margin is 2.17 times, at which point the mass m of the drug column is 497.63g, which meets the requirement of less than 1kg.

[0037] Based on the propellant grain design results for different pressure indices, propellants are divided into low-to-medium pressure index propellants (n≤0.38) and medium-to-high pressure index propellants (0.38<n≤0.55), and differentiated designs are implemented for the two types of propellants. For low-to-medium pressure index propellants, due to the small variation in burning rate with pressure and the low pressurization rate, the main consideration in the propellant grain design is to meet the pressure testing range. Therefore, a thin and long propellant grain is adopted, with a recommended grain length L≥150mm, nozzle throat diameter dt≤7.0mm, and wall thickness 15mm≤e≤20mm. For medium-to-high pressure index propellants, due to the large variation in burning rate with pressure and the high pressurization rate, the main consideration in the propellant grain design is to limit the pressurization rate from being too high; at the same time, considering the establishment of initial pressure, while reducing the propellant grain length, the inner diameter of the propellant grain should be appropriately increased. Therefore, a short and thick propellant grain is recommended, with a recommended grain length L < 150 mm, nozzle throat diameter dt > 7.0 mm, and inner diameter d ≥ 30 mm.

[0038] Using a short, thick propellant grain of the same size as the medium-high burning rate propellant 2 for propellant #1 (low-pressure index) results in an initial pressure of only 2.7 MPa, which is insufficient to reach the stable combustion pressure. Furthermore, due to the small burning surface area in the later stages of combustion, the final pressure is only 8.1 MPa, significantly lower than the design value, resulting in an insufficient testing range. If the medium-high burning rate propellant #2 uses the same slender propellant grain as propellant #1 (low-pressure index), the pressure testing range is 13.6 MPa-88.1 MPa, with a maximum pressurization rate reaching 429 MPa / s, exceeding the critical pressurization rate value. In this case, combustion cannot respond promptly to pressure changes; simultaneously, the combustion time is too short, only 0.39 s, which is unfavorable for burning rate testing. Therefore, the design approaches of the two propellants cannot be interchanged.

Claims

1. A method for designing a tubular propellant column for impulse-based burning rate testing, characterized in that, Includes the following steps: Step 1: Set the nozzle throat diameter The initial value; based on the nozzle throat diameter Calculate the propellant burning surface area ; Step 2: Calculate the minimum propellant burning surface area based on requirements. and obtain the lowest pressure. Set the initial value of the inner diameter d of the propellant column; Step 3: Calculate the propellant grain length based on the minimum propellant burning surface area. ; Minimum propellant burning surface area satisfy: ; according to Calculate the length of the propellant column ; If the length of the propellant column ≥200mm is considered If the diameter is too large, the inner diameter of the propellant grain in the tubular charge will be too small; in this case, the nozzle throat diameter dt will be reduced in increments of 0.2 mm, and the process will return to step 1 to recalculate the propellant burning surface area. If the length of the propellant column If the diameter is less than 200 mm, then the inner diameter d of the propellant column and the nozzle throat diameter are obtained. and the length of the medicine column ; Step 4: Determine the outer diameter D of the propellant column based on the range of the pressure index; Step 5: Based on the obtained outer diameter D, inner diameter d, and length of the propellant column... Calculate the pressurization rate of the tubular charge; if the pressurization rate is greater than or equal to 80% of the critical pressurization rate, it is considered that the pressurization rate is too high, so the inner diameter d of the propellant is reduced by a step size of 2mm, and the process jumps to step 3 to recalculate; if the pressurization rate is less than 80% of the critical pressurization rate, it is considered that the pressurization rate can meet the requirements for timely pressure response, and the inner diameter d of the propellant and the nozzle throat diameter are then... The length L and outer diameter D of the propellant charge are the design parameters for tubular propellant loading, which can meet the requirements of timely pressure response.

2. The method for designing a tubular propellant column for impulse-based burning rate testing according to claim 1, characterized in that, The specific process for calculating the boost rate in step 5 is as follows: Differentiating the transient equilibrium pressure formula in the combustion chamber with respect to time, we obtain the boost rate. They are represented as follows: （1） （2） In the formula, The rate of change of the burning surface, d is the inner diameter of the propellant column. Let be the thickness of the meat burned at time t, 0 ≤ ≤(Dd) / 2, where D is the outer diameter of the propellant column and L is the length of the propellant column; n is the burn rate coefficient; n is the pressure exponent; For propellant density; Characteristic velocity of the propellant; For the propellant burning surface area, This refers to the area of ​​the nozzle throat. That is (3) in This refers to the boost rate.

3. The method for designing a tubular propellant column for impulse-based burning rate testing according to claim 2, characterized in that, The step of obtaining the critical value of the boost rate is as follows: Step 5-1: Obtain the rate of change of propellant burning rate under the presence of pressurization rate; The rate of change of burning rate is the ratio of the difference between static pressure burning rate and dynamic pressure burning rate to the response time of dynamic pressure burning rate. The static pressure burning rate of the propellant under constant pressure is: The dynamic pressure burning rate corresponding to a rapid change in pressure is: Dynamic pressure combustion rate from Become The process takes time ,Right now The response time of dynamic pressure combustion rate; This is the combustion rate coefficient; Pressure; The static pressure burning rate of the propellant under constant pressure is: ; The dynamic pressure burning rate corresponding to rapid pressure change is: Dynamic pressure combustion rate from Become The process takes time ,Right now The response time of dynamic pressure combustion rate; The rate of change of combustion speed for: ； in n is the burn rate coefficient; n is the pressure exponent; It characterizes the degree of hysteresis in the response of dynamic pressure burning rate to pressure changes; Step S5-2: Rearrange the expression for the rate of change of burning rate to obtain the dynamic pressure burning rate and the dynamic relationship between the rate of change of burning rate and pressure. The rate of change of combustion rate By rearranging the terms, we obtain the dynamic relationship between dynamic pressure-induced combustion rate and the rate of change of combustion rate with pressure: The dynamic relationship between dynamic pressure-induced combustion rate and the rate of change of combustion rate with pressure is a classical first-order dynamic differential equation, while the dynamic relationship between dynamic pressure-induced combustion rate and the rate of change of combustion rate with pressure is a first-order inertial element; when the pressure is constant, i.e. At that time, the Vieri formula is equivalent to the dynamic relationship between dynamic pressure burning rate and the rate of change of burning rate and pressure. Step S5-3: Determine the boundary conditions for dynamic pressure burning rate and timely response to pressure changes; When the boost rate When it is large, because Since it is greater than zero, the dynamic relationship between dynamic pressure burning rate and the rate of change of burning rate and pressure is derived from the dynamic formula. Therefore, the dynamic pressure combustion rate at the boost rate is lower than the static pressure combustion rate at steady state, so it manifests as a decrease in the dynamic pressure combustion rate at the boost rate. When the pressure is high, the dynamic pressure combustion rate response will lag. To ensure that the dynamic pressure burning rate responds promptly to changes in pressure, the boundary conditions for the dynamic pressure burning rate to respond promptly to changes in pressure are determined as follows: When the boundary condition requirement of dynamic pressure burning rate responding to pressure change in a timely manner is met, the dynamic pressure burning rate can respond to pressure change in a timely manner. That is, the Vieri formula and the dynamic relationship between dynamic pressure burning rate and the rate of change of burning rate and pressure are equivalent. Step S5-4: Based on the boundary conditions of dynamic pressure combustion rate and timely response to pressure changes, differentiate the Vieri formula with respect to time to obtain the boundary conditions of the boost rate. Differentiating Vieri's formula with respect to time yields: Substituting the result obtained by differentiating the Vieri formula with respect to time into the boundary conditions for dynamic pressure-burning rate and timely response pressure changes, the boundary conditions for the boost rate are derived as follows: Step S5-5: Determine the critical value and range of boost rate; Take the product of the rate of change of burning rate and the dynamic pressure burning rate response time as less than the dynamic pressure burning rate. This is the critical condition, i.e. The critical value for obtaining the boost rate is calculated. ,Right now The pressurization rate range in which the propellant dynamic pressure burning rate responds promptly to pressure. Therefore, the critical value for boost rate is: Since the minimum time interval for the impulse method combustion rate test is 0.05s, the required combustion rate response time is... The time limit must not exceed 0.005 s. This study adopted... The baseline value is 0.005s, which means that the propellant burning rate is considered to respond promptly to pressure changes within 0.005s. This value ensures the applicability of the critical pressurization rate calculation and meets the burning rate testing requirements of most propellants. The unit of critical pressurization rate is MPa / s.

4. The method for designing a tubular propellant column for impulse-based burning rate testing according to claim 1, characterized in that, The nozzle throat diameter The initial value is 7mm.

5. The method for designing a tubular propellant column for impulse-based burning rate testing according to claim 1, characterized in that, The lowest pressure was obtained The steps are as follows: Minimum ignition rate Burn rate coefficient Pressure index n, propellant density Minimum value of propellant burning surface area nozzle throat area and propellant characteristic velocity Substituting the transient equilibrium pressure expression in the combustion chamber, the minimum pressure is obtained. Minimum pressure Lower than the minimum required pressure .

6. The method for designing a tubular propellant column for impulse-based burning rate testing according to claim 1, characterized in that, The step of determining the outer diameter D of the propellant column based on the range of the pressure index is as follows: The first candidate outer diameter is calculated based on the transient equilibrium pressure formula. ; When the pressure index n ≤ 0.38, calculate the second candidate outer diameter. for: ,in The first thickness is the thickest; to ensure sufficient burning time, the range of the first thickness is: ≥15mm; The outer diameter D of the propellant column is: ； When the pressure index is in the range of 0.38 < n ≤ 0.55, calculate the second candidate outer diameter. for: , This is the second thickest part of the meat; to ensure sufficient burning time, the range of the second thickest part is as follows: ≥20mm; The outer diameter D of the propellant column is: ； When the pressure index n>0.55, the tubular charge cannot meet the design requirements.

7. A terminal device, comprising a processor, a memory, and a computer program stored in the memory; characterized in that, When the processor executes the computer program, it implements the design method of the tubular propellant column for the impulse method burning rate test according to any one of claims 1-6.

8. A computer-readable storage medium storing a computer program; characterized in that, When the computer program is executed by the processor, it implements the design method of tubular propellant column for impulse method burning rate test according to any one of claims 1-6.

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