Design method of wedge-shaped grain for impulse method burning rate test based on pressurization rate boundary

Abstract

The invention discloses a design method of a wedge-shaped grain for testing a combustion rate by an impulse method based on a pressurization rate boundary. The problem that in an impulse method experiment, when an inner hole surface-increased combustion tubular grain is adopted for measuring the dynamic combustion speed of a solid propellant under different pressure intensities, the pressurization rate of a high-pressure-index propellant in the later stage of combustion is too high, and consequently the combustion speed cannot respond to pressure intensity changes in time is mainly solved. The method has the innovation points that (1) a mathematical relationship between the combustion surface area size and the change rate and the influence of the nozzle throat diameter on the pressurization rate is established, and a design method for controlling the pressurization process by regulating and controlling the combustion surface area is provided; (2) an outer wedge-shaped grain is designed on the basis of a traditional tubular grain, the combustion surface area and the change rate of the combustion surface area in the later stage of combustion are reduced, and the pressurization rate in the later stage of combustion is inhibited; and (3) in the design process of the wedge-shaped grain, the specific design process of the wedge angle of the wedge-shaped grain, the thickness of the cylinder section and the sizes of other grains is given in combination with the pressurization rate, and the design process of the wedge-shaped grain is standardized.

Description

Technical Field

[0001] This invention belongs to the field of solid propellant combustion performance testing technology, specifically relating to a wedge-shaped propellant grain design method for impulse burning rate testing based on pressurization rate boundary. Background Technology

[0002] Solid propellants, serving as the energy source and working fluid in solid rocket engines, are a crucial power source for missiles and rockets. The burning rate of a solid propellant, defined as the distance the solid phase disappears per unit time along the normal direction of the propellant's burning surface, is a key performance parameter in solid propellant formulation development and solid rocket engine design. Pressure is a major environmental factor affecting the burning rate of solid propellants; therefore, researching the combustion characteristics of solid propellants over a wide pressure range is an important prerequisite and foundation for solid rocket engine design.

[0003] In the impulse method for measuring the burning rate of solid propellants, a method was proposed that utilize the internal bore of the tubular propellant in engine testing to increase its surface area. The burning rate under different pressure ranges could be calculated using the thrust-time and pressure-time curves obtained from a single experiment. The document "Determination of the Validity of Raw Data from Impulse Method Solid Propellant Burning Rate Tests" introduces a method for determining ignition synchronization in impulse method burning rate testing. The premise of the impulse method is that combustion can respond instantaneously to pressure changes. However, when the pressurization rate is too high, combustion cannot respond instantaneously to pressure changes; therefore, controlling the pressurization rate is a crucial issue to consider in impulse method testing. This is especially true for high pressure index (n > 0.55) propellants, whose high burning rate and high pressure index make them more susceptible to pressure influence, and whose high propellant mass and large gas mass produced per unit time in the later stages of combustion, leading to excessively high pressurization rates. Summary of the Invention

[0004] To address the issue that in impulse burn rate testing, when using tubular propellant grains with internally vented combustion to measure the dynamic burn rate of solid propellants under different pressures, high-pressure index propellants may exhibit excessively high pressurization rates in the later stages of combustion, resulting in a burn rate that cannot respond promptly to pressure changes, this invention provides a wedge-shaped propellant grain design method for impulse burn rate testing based on pressurization rate boundaries.

[0005] According to the principle of solid rocket motors, the formula for the transient equilibrium pressure in the combustion chamber is: (1) The boost rate is obtained by differentiating the transient equilibrium pressure formula in the combustion chamber. for: (3) In the formula, P is the pressure; For combustion rate, ; This is the combustion rate coefficient; The pressure index; For propellant density, For the propellant burning surface area, This represents the area of ​​the nozzle throat. Characteristic velocity of the propellant; According to equations (1) and (3), the pressure P is related to the propellant burning surface area. Positive correlation, boost rate With propellant burning surface area and the rate of change of propellant combustion surface area over time Proportional; when the propellant combustion surface area and the rate of change of propellant combustion surface area over time When the pressure is reduced, the boost rate Reduced; in impulse burn rate tests, traditional tubular propellant grains burn with the flow of combustion, and the propellant burning surface area... The continuous increase leads to a rapid rise in pressure P. For high-pressure, high-exponential propellants with n > 0.55, the burning rate... Extremely sensitive to pressure changes, the burning rate response lags behind the pressure change when the pressure changes rapidly. According to equation (2), the pressurization rate can be limited by reducing the size of the propellant's burning surface area and the rate of increase of the burning surface area. It is required that the burning surface area of ​​the new propellant type remains unchanged in the early stage of combustion compared to the tubular propellant, while the size of the burning surface area and the rate of increase of the burning surface area are reduced in the later stage of combustion, thereby suppressing the pressurization rate. Therefore, an outer wedge-shaped propellant grain is proposed after geometrical adjustments based on the tubular propellant grain.

[0006] The design method for wedge-shaped propellant charges in impulse-based combustion rate testing based on pressurization rate boundaries includes the following steps: Step 1: Set the initial value of the nozzle throat diameter; Nozzle throat diameter The initial value is 7mm; Step 2: Based on the given minimum pressure The initial combustion surface is obtained by calculating the transient equilibrium pressure formula of the combustion chamber. Initial combustion surface The corresponding pressure is lower than the minimum pressure. ; The formula for the transient equilibrium pressure in the combustion chamber is: (1) Step 3: Set the initial value of the wedge-shaped charge inner diameter; To prevent erosion and combustion, the inner diameter of the wedge-shaped charge is initially set. It is 20mm; Initial burning surface for: ; Calculate the length of the wedge-shaped charge based on the initial burning surface. If L < 200mm, skip to step 4; If L is greater than or equal to 200mm, it indicates the inner diameter Too small, for nozzle throat diameter Correction: Gradually reduce the nozzle throat diameter in increments of 0.2 mm, then jump to step 1; Step 4: Calculate the outer diameter of the wedge-shaped charge. ; The propellant burning surface area in equation (1) of the transient equilibrium pressure formula in the combustion chamber Unfold to obtain the pressure when the burning surface is located in the cylindrical section: ...(2) The first candidate value of the outer diameter is obtained by calculating the pressure formula when the burning surface is located in the cylindrical section according to equation (2). ; Calculate the candidate value of the second outer diameter for:

[0007] When the pressure index n > 0.55, the wall thickness e must be greater than or equal to 20 mm while ensuring the burning time. Taking the wall thickness e = 20 mm, we obtain the candidate value for the second outer diameter. ; When the pressure index n <= 0.55, the wedge-shaped charge cannot meet the design requirements; outer diameter of wedge-shaped charge for: ; Step 5: Calculate the booster rate based on the designed drug formulation and determine the wedge angle. ; The boost rate is obtained by differentiating the transient equilibrium pressure formula in the combustion chamber. for: (3) The propellant burning surface area in equation (1) Unfold, and obtain the pressure when the burning surface advances to the frustum section: (4) Differentiating the transient equilibrium pressure in the combustion chamber with respect to time, the boost rate is... These are expressed as equations (5) and (6) respectively: Differentiating equation (2) with respect to time, we obtain equation (5) which shows the relationship between the boost rate of the cylindrical section and the combustion surface. (5) Differentiating equation (4) with respect to time, we obtain equation (6) which is the relationship between the truncated cone pressurization rate and the combustion surface. (6) Equation (5) represents the boost rate when the combustion surface is located in the cylindrical section; for high-pressure, high-exponential propellants, the boost rate increases with the current wall thickness. Rapid rise; for tubular charges, since only a cylindrical section exists, the change in pressurization rate during the entire combustion process is shown in equation (5). In the later stage of combustion, due to the current thickness of the charge... Rapid increases can easily lead to excessive boost rate. Equation (6) represents the rate of pressure increase when the combustion surface advances to the frustum section. Since the pressure is required to rise continuously during the test, the combustion area must continue to increase, and the wedge angle... The following formula must be satisfied: (7) For a pressure index n > 0.55, the pressure and pressurization rate in the later stages of combustion will be much greater than the critical value; according to equation (5), the pressurization rate versus time curve of the tubular charge is calculated. In order to retain a certain pressurization rate margin, when the pressurization rate is equal to half of the critical value of the pressurization rate, that is... At that time, determine the current moment. t 1 represents the moment when the cylindrical section finishes burning and the frustum-shaped section begins burning; according to A b -t Curve determination t At time 1, the thickness of the cylindrical segment Based on the thickness of the cylindrical section Determine the thickness of the frustum section. L y ;in This is the critical value for the boost rate; Set wedge angle The initial value is 90°; The pressure and pressurization rate are calculated using equations (4) and (6). If equations (4) and (6) simultaneously satisfy conditions A and B, then the wedge angle is... The requirement is met; if either condition A or condition B is not met, then the wedge angle... Decrease the nozzle throat diameter in increments of 10°, while simultaneously decreasing it in increments of 0.2mm. Calculate the corresponding pressure using the updated nozzle throat diameter, and repeat step 5. Condition A: The pressure is greater than the maximum pressure required for testing; Condition B: The boost rate is less than 80% of the critical boost rate value; The final wedge angle obtained , inner diameter , outer diameter Length of the propellant column Nozzle throat diameter The cylindrical section has thick flesh. This refers to the final size of the wedge-shaped propellant column.

[0008] Furthermore, the steps for determining the critical value of the boost rate are 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 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.

[0009] 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:

[0010] 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 is equivalent to the dynamic pressure burning rate and the dynamic relationship between the rate of change of burning rate and pressure. 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:

[0011] 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 condition for the boost rate is obtained as follows:

[0012] 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. Therefore, the critical value for boost rate is:

[0013] 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.

[0014] The beneficial effects of this invention are: (1) Quantitatively reveal the synergistic effect mechanism of combustion surface area and nozzle throat diameter on pressurization rate: By quantitatively calculating the pressurization rate, the quantitative influence of combustion surface area and combustion surface area change rate on pressurization rate is demonstrated, providing a theoretical basis for subsequent propellant design.

[0015] (2) In response to the problem of excessively high pressurization rate that is prone to occur in high-pressure high-index propellants, while demonstrating that tubular charge is not feasible, an external wedge-shaped propellant grain structure design is proposed. Its special geometry is used to limit the rate of increase of the burning surface area in the later stage of combustion, thereby reducing the pressurization rate.

[0016] (3) A quantitative relationship model between the size of the wedge-shaped propellant and the pressure and pressurization rate was established, the design steps of the size of the wedge-shaped propellant were standardized, and a design method for the size of the propellant, including the wedge angle and the thickness of the cylindrical section, was proposed. The control of the change rate of the burning surface area was realized, and a feasible propellant design scheme was provided for measuring the burning rate of high pressure index propellants by the impulse method. Attached Figure Description

[0017] Figure 1 Schematic diagram of a tubular propellant grain; Figure 2 Schematic diagram of a wedge-shaped propellant grain; Figure 3 The graph shows the changes in pressure and burning surface area of ​​a tubular propellant charge over time. Figure 4 The curves showing the variation of the burning surface area of ​​the wedge-shaped propellant charge with meat thickness at different angles; Figure 5 This is a drawing showing the dimensions of the designed wedge-shaped propellant column; Figure 6 To design the Pt vs. Ft curves of the rear wedge-shaped propellant column; Wherein, 1 is the coating layer; 2 is the propellant; and 3 is the initial combustion surface. Detailed Implementation

[0018] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0019] According to the principle of solid rocket motors, the formula for the transient equilibrium pressure in the combustion chamber is: (1) The boost rate is obtained by differentiating the transient equilibrium pressure formula in the combustion chamber. for: (3) In the formula, P is the pressure; For combustion rate, ; This is the combustion rate coefficient; The pressure index; For propellant density, For the propellant burning surface area, This represents the area of ​​the nozzle throat. Characteristic velocity of the propellant; According to equations (1) and (3), the pressure P is related to the propellant burning surface area. Positive correlation, boost rate With propellant burning surface area and the rate of change of propellant combustion surface area over time Proportional; when the propellant combustion surface area and the rate of change of propellant combustion surface area over time When the pressure is reduced, the boost rate Reduced; in impulse burn rate tests, traditional tubular propellant grains burn with the flow of combustion, and the propellant burning surface area... The continuous increase leads to a rapid rise in pressure P. For high-pressure, high-exponential propellants with n > 0.55, the burning rate... It is extremely sensitive to pressure changes, and the burning rate response lags behind the pressure change. According to equation (3), the pressurization rate is limited by reducing the size of the propellant burning surface area and the rate of burning surface rise. It is required that the burning surface area of ​​the new propellant type remains unchanged in the early stage of combustion compared to the tubular charge, and that the size of the burning surface area and the rate of burning surface rise decrease in the later stage of combustion, thereby suppressing the pressurization rate. Therefore, an outer wedge-shaped propellant grain is proposed after geometrical adjustment based on the tubular propellant grain.

[0020] The design method for wedge-shaped propellant charges in impulse-based combustion rate testing based on pressurization rate boundaries includes the following steps: Step 1: Set the initial value of the nozzle throat diameter; 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. Step 2: Based on the given minimum pressure The initial combustion surface is obtained by calculating the transient equilibrium pressure formula of the combustion chamber. Initial combustion surface The corresponding pressure is lower than the minimum pressure. ; Minimum ignition rate Burn rate coefficient Pressure index n, propellant density and the minimum area of ​​the propellant combustion surface nozzle throat area Substituting the propellant characteristic velocity C* into the transient equilibrium pressure formula (1) in the combustion chamber, the minimum pressure is obtained. The nozzle throat area is obtained by calculating the nozzle throat diameter. Step 3: Set the initial value of the wedge-shaped charge inner diameter; To prevent erosion and combustion, the inner diameter of the wedge-shaped charge is initially set. 20mm; initial flammability for:

[0021] Calculate the length of the wedge-shaped charge based on the initial burning surface. If L < 200mm, skip to step 4; If L is greater than or equal to 200mm, it indicates the inner diameter Too small, for nozzle throat diameter Correction: Gradually reduce the nozzle throat diameter in increments of 0.2 mm, and repeat steps 1 to 3. Step 4: Calculate the outer diameter of the wedge-shaped charge. ; The propellant burning surface area in equation (1) of the transient equilibrium pressure formula in the combustion chamber Unfold to obtain the pressure when the burning surface is located in the cylindrical section: ...(2) The first candidate value of the outer diameter is obtained by calculating the pressure formula when the burning surface is located in the cylindrical section according to equation (2). ; Calculate the candidate value of the second outer diameter for:

[0022] When the pressure index n > 0.55, the wall thickness e must be greater than or equal to 20 mm while ensuring the burning time. Taking the wall thickness e = 20 mm, we obtain the candidate value for the second outer diameter. ; When the pressure index n <= 0.55, the wedge-shaped charge cannot meet the design requirements; outer diameter of wedge-shaped charge for: ; Step 5: Calculate the booster rate based on the designed drug formulation and determine the wedge angle. ; The propellant combustion surface area in Equation 2 Unfold, and obtain the pressure when the burning surface advances to the frustum section: (4) Differentiating the transient equilibrium pressure in the combustion chamber with respect to time, the boost rate is... These are expressed as equations (5) and (6) respectively: Differentiating equation (2) with respect to time, we obtain equation (5) which shows the relationship between the boost rate of the cylindrical section and the combustion surface. (5) Differentiating equation (4) with respect to time, we obtain equation (6) which is the relationship between the truncated cone pressurization rate and the combustion surface. (6) Equation (5) represents the boost rate when the combustion surface is located in the cylindrical section; for high-pressure, high-exponential propellants, the boost rate increases with the current wall thickness. Rapid rise; for tubular charges, since only a cylindrical section exists, the change in pressurization rate during the entire combustion process is shown in equation (5). In the later stage of combustion, due to the current thickness of the charge... Rapid increases can easily lead to excessive boost rate. Equation (6) represents the rate of pressure increase when the combustion surface advances to the frustum section. Since the pressure is required to rise continuously during the test, the combustion area must continue to increase, and the wedge angle... The following formula must be satisfied: (7) The wedge-shaped propellant determined by equation (7) avoids pressure reduction while limiting the pressurization rate in the later stages of combustion; for large wedge angles The combustion process in the frustum section exhibits a pressure curve similar to that of a cylindrical section, making it suitable for propellants with high pressure indices; for small wedge angles... Because the rate of increase in burning surface area decreases rapidly in the later stages of combustion, the rate of decrease in pressurization is significantly less than that of tubular propellants in the later stages of combustion, making it suitable for propellants with high pressure index.

[0023] For a pressure index n > 0.55, the pressure and pressurization rate in the later stages of combustion will be much greater than the critical value. The pressurization rate versus time curve for tubular charges is calculated according to equation (5). To retain a certain margin in pressurization rate, when the pressurization rate is equal to half of the critical value of the pressurization rate, i.e. At that time, determine the current moment. t 1 represents the moment when the cylindrical section finishes burning and the frustum-shaped section begins burning; according to A b-t Curve determination t At time 1, the thickness of the cylindrical segment ; Based on the thickness of the cylindrical section Determine the thickness of the frustum section. L y ; Set wedge angle The initial value is 90°; the pressure and pressurization rate are calculated using equations (4) and (6). If both conditions A and B are met, then the wedge angle is... The requirements are met; if any condition is not met, then the wedge angle... Decrease the nozzle throat diameter in increments of 10°, while simultaneously decreasing it in increments of 0.2mm. Calculate the corresponding pressure using the updated nozzle throat diameter, and repeat step 5. A. The pressure is greater than the maximum pressure required for testing; B. The boost rate is less than 80% of the critical boost rate value; The final wedge angle obtained , inner diameter , outer diameter Length of the propellant column Nozzle throat diameter The cylindrical section has thick flesh. This refers to the final dimensions of the wedge-shaped propellant grain; The steps for determining 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:

[0024] 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; 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 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:

[0025] 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:

[0026] 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 is equivalent to the dynamic pressure burning rate and the dynamic relationship between the rate of change of burning rate and pressure. 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:

[0027] 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 condition for the boost rate is obtained as follows:

[0028] 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:

[0029] 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. The specific process for designing wedge-shaped propellant grains is as follows: (1) Under normal circumstances, the propellant research and production unit will provide the density of the propellant to be tested ( ρ Characteristic velocity () C* Theoretical specific impulse and combustion rate at 7 MPa (or 6.86 MPa) r ), pressure index ( n The approximate range of the propellant grain's burning surface A at the initial moment of combustion. b0 It can be represented as:

[0030] In the formula, A b0 This is the initial ignition surface of the propellant column; P startUse 5-7 MPa (the specific pressure depends on the experimental requirements); r 0 is the propellant in P start The combustion rate is reduced.

[0031] Depending on the type of propellant and the test pressure range, the initial pressure... P start The range also varies; based on design experience, the nozzle throat diameter ( d t The initial diameter is 7mm. Since the propellant grain is an internally combusting tubular charge, the initial combustion surface is cylindrical. Calculations can determine the initial combustion surface diameter of the propellant grain. A b0 During the experiment, the pressure range must cover the required pressure test range, therefore the initial burning surface... A b0 The corresponding pressure should be slightly lower than the required initial pressure. P start Based on erosion and combustion suppression criteria and design experience, the inner diameter... Set to 20mm, initial burning surface meets requirements. Therefore, the length of the propellant charge can be calculated based on the initial ignition surface. , L Must meet L <200mm, when L If it is too large, it indicates that the inner diameter is too large. If it is too small, the inner diameter should be increased appropriately. .

[0032] (2) To ensure a certain burning time, take the thickest part of the meat. e =20mm. Based on the required maximum pressure. P The outer diameter calculated by max D 1 and based on meat thickness e =20mm outer diameter calculated D 2. Take the larger value as the outer diameter. .

[0033] For high-pressure, high-index propellants ( n >0.55) The pressure and pressurization rate in the later stages of combustion will be much greater than the critical value. The pressure of the tubular charge is calculated according to equation (2). Pt The curve and the relationship between the burning surface and time ( Ab-t )curve.

[0034] To maintain a certain boost rate margin, the boost rate should be equal to half of the critical boost rate. At that time, determine the current moment. t 1 represents the moment when combustion ends in the cylindrical section and begins in the frustum-shaped section. According to... A b -tCurve determination t Meat thickness at time 1 e 1.

[0035] (3) Given the thickness of the cylindrical section The thickness of the frustum section can be determined. L y By repeatedly adjusting the wedge angle Adjust the pressure and pressurization rate in the later stages of combustion. Considering the manufacturing process, the wedge angle... Try to set it to a multiple of 10. Wedge angle Decrease in increments of 10°, as... α As the pressure decreases, both the pressure and the boost rate decrease simultaneously. If the pressure test range is reasonable but the boost rate is still too high, then reducing the nozzle throat diameter is the appropriate option. Increase the pressure, then adjust the wedge angle. Further reductions will be implemented.

[0036] In a preferred embodiment, the inner diameter Satisfying 20mm≤ d ≤30mm: d Erosion and burning are likely to occur when the thickness is less than 20mm. d A diameter greater than 30mm will result in an excessively large volume and mass of the propellant column.

[0037] In a preferred embodiment, the nozzle throat diameter Satisfying 6mm < d t ≤15mm. When the nozzle throat diameter is too small, the pressure change rate during the test will be too large, resulting in inaccurate combustion rate results. In severe cases, it may lead to excessive combustion chamber pressure and engine explosion, and may also cause nozzle blockage. When the nozzle throat diameter is too large, the maximum pressure will be reduced, failing to meet the test requirements. At the same time, it will increase the amount of propellant used in a single test, increasing the test cost and unnecessary waste.

[0038] In a preferred embodiment, the fitted result Pt The curve graph can effectively predict the actual combustion yield. Pt Line graph.

[0039] In a preferred embodiment, wedge-shaped propellant grains can significantly reduce the combustion area and reduce the pressurization rate in the later stages of combustion compared to tubular propellant grains.

[0040] One embodiment of the present invention is as follows: The propellant selected is a certain type of low-emissivity propellant with a known density. ρ It is 1.74 g / cm³ 3 Characteristic velocity C* The combustion rate is 1480 m / s at 6 MPa. rThe pressure index is 7.331 mm / s. n The value is 0.72; the required pressure test range is 6-30 MPa.

[0041] The propellant type was designed according to the specific process of wedge-shaped propellant grain design.

[0042] (1) First, the initial burning surface size is calculated, and the nozzle throat diameter is initially taken as 7 mm, and the initial pressure is... P start Take 6MPa, and use the formula The initial flammability can be calculated. A b0 12235mm 2 Assuming the inner diameter of the propellant column... It is 20mm, according to the initial burning surface formula. The length of the propellant can be calculated. It is 194.8 mm. At this point, the propellant column length (L) is close to the critical value of 200 mm. L The diameter of the propellant column is too large, therefore the inner diameter of the propellant column needs to be appropriately increased. shorten the length L Inner diameter When expanded to 30mm, it becomes 12235mm. 2 initial burning surface A b0 Corresponding column length The length is 129.9 mm. Since the initial pressure needs to be lower than 6 MPa, the length of the propellant column is taken as 129.9 mm. It is 100mm.

[0043] (2) The maximum pressure required for testing is 30 MPa, corresponding to the maximum burning surface. A bmax 19201mm 2 The outer diameter was calculated. D 1 is 61.1mm, based on the wall thickness. e =20mm outer diameter calculated D 2 is 70mm. The larger value is taken, therefore the outer diameter is determined to be 70mm. At this point, the internal ballistic curve is calculated for the determined tubular charge, as follows... Figure 3 As shown, the pressure in the later stages of combustion was found to be much higher than the experimental requirements, and the maximum pressurization rate in the later stages of combustion reached 163.8 MPa / s, exceeding the critical pressurization rate. At a pressure of 135.1 MPa / s, the pressurization rate exceeds the critical pressurization rate, and the burning rate will not respond promptly to pressure changes. Therefore, tubular charges cannot meet the requirements of the high pressure index propellant impulse method test. Analysis shows that at a combustion time of 0.64 s, the combustion pressure is 4.55 MPa, and the actual pressurization rate is 5.41 MPa / s. The critical pressurization rate is 12.64 MPa / s, meaning the actual pressurization rate is approaching 50% of the critical pressurization rate. This corresponds to a wall thickness of 3 mm; therefore, the wall thickness of the cylindrical section is determined to be 3 mm.

[0044] (3) The wedge-shaped propellant column is defined as having a length of 100 mm, a cylindrical section thickness of 3 mm, and a frustoconical section thickness of 17 mm. At this point, different wedge angles... The variation of the lower combustion surface with the thickness of the wall is as follows: Figure 4 As shown, the wedge angle can be calculated according to equation (7). The critical value is 27.4°. α The value range is 27.4° < α <90°, therefore the wedge angle The pressure range is calculated sequentially from 90° in increments of 10° according to equation (4), and the pressurization rate is calculated according to equation (6). The wedge angle is then determined when the pressure test range of 6-30 MPa is met. At a wedge angle of 70°, the pressure range is 2.36-38.82 MPa, but the boost rate is only 94.31 MPa / s, while the critical boost rate is 107.83 MPa / s. The difference between these two values ​​is small, indicating a risk that the combustion rate may not respond promptly to pressure changes. To reduce the boost rate, the wedge angle needs to be further decreased. However, this reduces the pressure testing range, and the initial pressure is too low, which is detrimental to the ignition of the propellant. To address this, the nozzle throat diameter needs to be reduced. When the nozzle throat diameter When the pressure is reduced to 6.2 mm, the initial pressure reaches 5.62 MPa. At this point, calculations are continued at different wedge angles with a gradient of 10°. The pressure range and pressurization rate under different wedge angles At 30°, the pressure test range is 5.62MPa-33.39MPa, which covers the required pressure test range. The highest pressurization rate occurs at 0.90s of combustion time, at which point the combustion pressure is 24.78MPa and the pressurization rate is 30.46MPa / s, far below the critical pressurization rate of 68.83MPa / s. This reduction is mainly attributed to the fact that the geometric design of the wedge-shaped propellant grain effectively suppresses the size of the burning surface area at the end of combustion, thus significantly reducing the pressurization rate. At this point, the propellant grain is like... Figure 5 As shown, the internal ballistic curve is as follows Figure 6As shown. Therefore, the wedge-shaped propellant grain structure can reduce the pressurization rate while ensuring the pressure test range, ensuring that the propellant burning rate can respond promptly to pressure changes in the impulse method burning rate test, improving the reliability of the burning rate test, and more accurately reflecting the combustion performance of the propellant in the actual working environment.

Claims

1. A wedge-shaped propellant charge design method for impulse-based burning rate testing based on boost rate boundary, characterized in that, Includes the following steps: Step 1: Set the initial value of the nozzle throat diameter; Step 2: Based on the given minimum pressure The initial combustion surface is obtained by calculating the transient equilibrium pressure formula of the combustion chamber. Initial combustion surface The corresponding pressure is lower than the minimum pressure. ; Step 3: Set the initial value of the inner diameter of the wedge charge, and calculate the length of the wedge charge based on the initial burning surface; Initial setting of the inner diameter of the wedge-shaped charge 20mm; initial flammability for: ; Calculate the length of the wedge-shaped charge based on the initial burning surface. If L < 200mm, skip to step 4; If L is greater than or equal to 200mm, it indicates the inner diameter Too small, for nozzle throat diameter Correction: Gradually reduce the nozzle throat diameter in increments of 0.2 mm, then jump to step 1; Step 4: Calculate the outer diameter of the wedge-shaped charge. ; Step 5: Calculate the boost rate and determine the wedge angle. ; The boost rate is obtained by differentiating the transient equilibrium pressure formula in the combustion chamber. for: ； in This is the combustion rate coefficient; The pressure index; For propellant density, For the propellant burning surface area, This represents the area of ​​the nozzle throat. Characteristic velocity of the propellant; The propellant burning surface area in the formula for transient equilibrium pressure in the combustion chamber Unfold, and obtain the pressure when the burning surface advances to the frustum section: ； Differentiating the pressure formula with respect to time when the combustion surface is in the cylindrical section, we obtain the relationship between the pressurization rate in the cylindrical section and the combustion surface: ； By differentiating the pressure with respect to time when the combustion surface is advanced to the frustum section, the relationship between the frustum pressurization rate and the combustion surface is obtained: ； Because the pressure continuously increases during the test, the combustion area must continuously increase, and the wedge angle... Satisfy the following formula: ； For a pressure index n > 0.55, the pressure and pressurization rate in the later stages of combustion will be much greater than the critical value. Based on the relationship between the pressurization rate of the cylindrical section and the combustion surface, the pressurization rate versus time curve for the tubular charge is calculated. To retain a certain pressurization rate margin, when the pressurization rate equals half of the critical pressurization rate, i.e. At that time, determine the current moment. t 1 represents the moment when the cylindrical section finishes burning and the frustum-shaped section begins burning; according to A b -t Curve determination t At time 1, the thickness of the cylindrical segment Determine the thickness of the frustum section based on the thickness of the cylindrical section. L y ;in This is the critical value for the boost rate; Set wedge angle The initial value is 90°; The pressure and pressurization rate are calculated using the relationship between the pressure and the pressurization rate of the frustum when the combustion surface advances to the frustum section and the combustion surface. If the relationship between the pressure and the pressurization rate of the frustum when the combustion surface advances to the frustum section and the combustion surface simultaneously satisfies conditions A and B, then the wedge angle... The requirement is met; if either condition A or condition B is not met, then the wedge angle... Decrease the nozzle throat diameter in increments of 10°, while simultaneously decreasing it in increments of 0.2 mm. Calculate the corresponding pressure using the updated nozzle throat diameter, and repeat step 5. The final wedge angle obtained , inner diameter , outer diameter Length of the propellant column Nozzle throat diameter The cylindrical section has thick flesh. This refers to the final size of the wedge-shaped propellant column.

2. The wedge-shaped propellant charge design method for impulse-based combustion rate testing based on the boost rate boundary as described in claim 1, characterized in that, The steps for determining the critical value of the boost rate are as follows: Step 5-1: Obtain the rate of change of propellant burning rate under 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 is equivalent to the dynamic pressure burning rate and the dynamic relationship between the rate of change of burning rate and pressure. 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 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. 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 seconds. The baseline value is 0.005s, which means that the propellant burning rate is considered to respond promptly to pressure changes within 0.005s.

3. The wedge-shaped propellant charge design method for impulse-based burning rate testing based on pressurization rate boundary as described in claim 1, characterized in that, The nozzle throat diameter The initial value is 7mm.

4. The wedge-shaped propellant charge design method for impulse-based combustion rate testing based on the boost rate boundary as described in claim 1, characterized in that, The formula for the transient equilibrium pressure in the combustion chamber is: （1） Wherein, P is the pressure; For combustion rate, ; This is the combustion rate coefficient; The pressure index; For propellant density, For the propellant burning surface area, This represents the area of ​​the nozzle throat. The characteristic velocity of the propellant.

5. The wedge-shaped propellant charge design method for impulse-based combustion rate testing based on pressurization rate boundary as described in claim 1, characterized in that, Calculate the outer diameter of the wedge-shaped charge The steps are as follows: The propellant burning surface area in the transient equilibrium pressure formula in the combustion chamber Unfold to obtain the pressure when the burning surface is located in the cylindrical section: ……（2） The first candidate value of the outer diameter is obtained by calculating the pressure formula when the burning surface is located in the cylindrical section according to equation (2). ; Calculate the candidate value of the second outer diameter for: When the pressure index n > 0.55, and to ensure the combustion time, the wall thickness e must be greater than or equal to 20 mm to obtain the second candidate value for the outer diameter. ; When the pressure index n <= 0.55, the wedge-shaped charge cannot meet the design requirements; outer diameter of wedge-shaped charge for: ;in To take the maximum value of the first outer diameter candidate value and the first outer diameter candidate value.

6. The wedge-shaped propellant charge design method for impulse-based combustion rate testing based on pressurization rate boundary as described in claim 1, characterized in that, Conditions A and B are respectively: Condition A: The pressure is greater than the maximum pressure required for testing; Condition B: The boost rate is less than 80% of the critical boost rate value.

7. A computer-readable storage medium storing a computer program; characterized in that, When the computer program is executed by the processor, it implements the wedge-shaped propellant design method for impulse combustion rate testing based on the boost rate boundary as described in any one of claims 1-6.

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