Method for determining the boundary condition of pressurization rate in impulse method of testing burning rate

Abstract

The present application provides a method for determining the boundary condition of the pressurization rate in the impulse method of testing the burning rate; it belongs to the technical field of testing the combustion performance of solid propellant; the present application defines the pressure burning rate as the burning rate of propellant under a certain constant pressure, which is only related to the pressure; when the dynamic pressure burning rate changes rapidly, the burning rate depends not only on the current pressure, but also on the pressure change rate; the present application starts from the “burning rate lag caused by pressure change”, and obtains that when the product of the burning rate change rate and the response time is much smaller than the burning rate under the current constant pressure, it is considered that “the increasing pressure of propellant combustion is instantaneously responsive”; according to the burning rate response time, a pressurization rate relational expression containing a time term is established, and when the burning rate response time is less than 5 ms, the expression of the pressurization rate limiting value of the burning rate timely responding to the pressure change is given, which is used for judging the effectiveness of the pressure curve in the impulse method of testing the burning rate, that is, the effectiveness of the original data.

Description

Technical Field

[0001] This invention belongs to the field of solid propellant combustion performance testing technology, specifically relating to a method for determining the pressurization rate boundary conditions in impulse burning rate testing. Background Technology

[0002] The burning rate of solid propellants under different pressures is a crucial parameter characterizing their combustion properties, a fundamental parameter in rocket engine design, and an important parameter for predicting engine ballistic performance. Burning rate is the distance the propellant charge's burning surface recedes along its normal direction per unit time; it is simply called the solid propellant burning rate. It not only directly determines the rate of energy release of solid propellants but is also a core parameter for calculating other combustion properties of solid propellants (burning rate coefficient, burning rate pressure index, burning rate temperature sensitivity coefficient, erosion ratio, etc.).

[0003] In impulse method tests, using a propellant grain with a specific configuration and increased surface area, the burning rate at any pressure within the pressurization range can be obtained from the thrust-time and pressure-time curves measured in a single experiment. The patent "Validity Judgment of Original Data for Impulse Method Solid Propellant Burning Rate Test" (ZL202110283176.2, 2022.3) proposes a criterion for determining the ignition synchronization of the exposed propellant burning surface. At excessively high pressurization rates, the burning rate exhibits a lag with pressure changes; therefore, the impulse method requires the propellant combustion energy to respond instantaneously to pressure increases. In existing technologies, the criteria for determining the critical pressurization rate in impulse method burning rate tests remain undefined, potentially leading to errors in the prediction of the internal ballistic curve. Furthermore, the design of the propellant grain size lacks clear constraints in this regard. Summary of the Invention

[0004] In order to overcome the shortcomings of the existing technology and solve the problem that the critical boost rate cannot be determined in the impulse method combustion rate test, the present invention provides a method for determining the boost rate boundary condition in the impulse method combustion rate test.

[0005] The technical solution adopted by this invention to solve its technical problem includes the following steps:

[0006] Step S1: Obtain the rate of change of propellant burning rate under the presence of pressurization rate;

[0007] 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;

[0008] Step S2: 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] Step S3: Determine the boundary conditions for dynamic pressure burning rate and timely response to pressure changes;

[0010] Step S4: 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.

[0011] Step S5: Determine the critical value and range of boost rate.

[0012] Furthermore, the specific steps for obtaining the rate of change of propellant burning rate under a pressurization rate are as follows:

[0013] The static pressure burning rate of the propellant under constant pressure is: ;

[0014] 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;

[0015] The rate of change of burning rate is the ratio of the difference between the static pressure burning rate and the dynamic pressure burning rate to the response time, i.e., the rate of change of burning rate. for:

[0016] ; (2)

[0017] in n is the burn rate coefficient; n is the pressure exponent. It characterizes the degree of lag in the response of dynamic pressure burning rate to pressure changes.

[0018] Furthermore, the step of obtaining the dynamic pressure-burning rate and the dynamic relationship between the rate of change of burning rate and pressure is as follows:

[0019] By rearranging the terms in equation (2), we obtain the dynamic relationship between dynamic pressure burning rate and the rate of change of burning rate and pressure (3).

[0020] (3)

[0021] Equation (3) is a classical first-order dynamic differential equation. The dynamic relationship between dynamic pressure, combustion rate, and 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 equation (3).

[0022] Furthermore, the boundary conditions for determining the dynamic pressure burning rate and responding promptly to pressure changes;

[0023] When boost rate When it is large, because It is greater than zero, so according to equation (3), we get 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.

[0024] 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:

[0025] (4)

[0026] When the requirement of equation (4) is met, the dynamic pressure burning rate can respond to the change of pressure in a timely manner, that is, the Vieri formula is equivalent to equation (3).

[0027] Furthermore, the step of obtaining the boundary conditions for the boost rate is as follows:

[0028] Differentiating Vieri's formula with respect to time yields:

[0029] (5)

[0030] Substituting equation (5) into equation (4), the boundary condition for the boost rate is obtained as follows:

[0031] (6)

[0032] in This refers to the boost rate.

[0033] Furthermore, the steps for determining the critical value and range of the boost rate are as follows:

[0034] 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.

[0035] The critical value for boost rate is:

[0036] ; (7)

[0037] in, It is 0.005 s.

[0038] The beneficial effects of this invention are as follows: This invention establishes the relationship between burning rate and pressure under dynamic pressure, proposes the boundary conditions for pressurization rate in the impulse method test process, and proposes a method for determining the critical value of pressurization rate for the "instantaneous response of propellant combustion to pressure increase," thus improving the theoretical basis of the impulse method burning rate test technology and ensuring the consistency and validity of the test results. This invention solves the problem of the inability to determine the critical pressurization rate in the impulse method burning rate test, explains the problem of the burning rate being too low when the pressurization rate is too high in the impulse method burning rate test, and makes the burning rate test value more accurate. This improves the rationality of propellant structure design in engineering practice, makes the prediction of internal ballistic curves more accurate, and is conducive to promoting the application of solid propellants in actual production. Attached Figure Description

[0039] Figure 1 The Pt and Ft curves obtained by the impulse method test are shown below; (a) shows the Pt and Ft curves of the test results with a large boost rate; (b) shows the Pt and Ft curves of the test results with a small boost rate.

[0040] Figure 2 The pressurization rate-pressure curves are shown for the two sets of experiments.

[0041] Figure 3 The combustion rate-pressure curves for the two sets of tests are shown.

[0042] Figure 4 The graphs are segmented lnr-lnP curves for composite propellant 1; where (a) is the curve from 5.61 MPa to 32.44 MPa; and (b) is the curve from 32.44 MPa to 74.96 MPa.

[0043] Figure 5 The graphs are segmented lnr-lnP curves for composite propellant 2 under different pressures; where (a) is the curve for 5.83~9.39MPa; (b) is the curve for 9.39~20.18MPa; and (c) is the curve for 20.18MPa~32.62MPa. Detailed Implementation

[0044] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0045] The method for determining the boost rate boundary conditions in the impulse-based combustion rate test is as follows:

[0046] For solid propellants, the relationship between the burning rate and pressure under static pressure is expressed by the Vieri formula:

[0047] (1)

[0048] 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.

[0049] 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.

[0050] Step S1: Obtain the rate of change of propellant burning rate under the presence of pressurization rate;

[0051] 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.

[0052] The static pressure burning rate of the propellant under constant pressure is: ;

[0053] 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;

[0054] The rate of change of combustion speed for:

[0055] ; (2)

[0056] 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;

[0057] Step S2: 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.

[0058] By rearranging the terms in equation (2), we obtain the dynamic relationship between dynamic pressure burning rate and the rate of change of burning rate and pressure (3).

[0059] (3)

[0060] Equation (3) is a classical first-order dynamic differential equation. The dynamic relationship between dynamic pressure, combustion rate, and 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 equation (3);

[0061] Step S3: Determine the boundary conditions for dynamic pressure burning rate and timely response to pressure changes;

[0062] When boost rate When it is large, because It is greater than zero, so according to equation (3), we get 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.

[0063] 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:

[0064] (4)

[0065] When the requirement of equation (4) is met, the dynamic pressure burning rate can respond to the change of pressure in a timely manner, that is, the Vieri formula is equivalent to equation (3);

[0066] Step S4: 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.

[0067] Differentiating Vieri's formula with respect to time yields:

[0068] (5)

[0069] Substituting equation (5) into equation (4), the boundary condition for the boost rate is obtained as follows:

[0070] (6)

[0071] Step S5: Determine the critical value and range of boost rate;

[0072] 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.

[0073] 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, for dynamic pressure burning rate, timely response to pressure change boundary conditions (4) can satisfy In signal processing theory, minor components in a signal can be ignored when their proportion is less than 1%. For equation (4), take... 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%, and for equation (4) we take 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.

[0074] Therefore, the critical value for boost rate is:

[0075] (7)

[0076] 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 set at 0.005 s, which means that the propellant burning rate is considered to respond promptly to pressure changes within 0.005 s. 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.

[0077] As known from Equation (7), the higher the value of n, the smaller the critical value of the pressure boosting rate. This is because the pressure exponent n reflects the sensitivity of the burning rate r to the change in pressure P. The larger the value of n, the greater the amplitude of the change in the burning rate with pressure fluctuations, and the smaller the rate of change in pressure that the burning rate can respond to within a specified response time. That is, the critical value of the pressure boosting rate is negatively correlated with the pressure exponent n.

[0078] The influence of pressure P on the pressure boosting rate : According to the Vieille formula, taking the derivative of the pressure P gives the rate of change of the burning rate:

[0079] (8)

[0080] Since 0 < n < 1, then n - 1 < 0, indicating that as the pressure P increases, the rate of change of the burning rate with pressure gradually decreases, and the sensitivity of the burning rate to pressure fluctuations decreases. Therefore, the pressure boosting rate is positively correlated with the instantaneous pressure P.

[0081] Based on the above analysis, when the pressure exponent n is low and the pressure P is high, the critical pressure boosting rate of the propellant combustion system is relatively high, and the combustion stability is better, which is consistent with the calculation results of Equation (7).

[0082] The general process for measuring the burning rate of a certain hydroxyl-terminated polybutadiene (HTPB) three-component composite propellant by the impulse method is as follows:

[0083] (1) Measure the dimensions of the tubular propellant grain: Composite propellant 1: outer diameter (D) 55 mm, inner diameter (d) 20 mm, length (L) 210 mm, mass (m) 777 g; Composite propellant 2: outer diameter (D) 55 mm, inner diameter (d) 20 mm, length (L) 179 mm, mass (m) 664.5 g. Coat the outer surface and end faces of the grain, leaving only the inner surface as the initial burning surface.

[0084] (2) Select the nozzle throat diameter. Freely load the grain into the combustion chamber, connect the ignition circuit and sensors (pressure, thrust). Ignite and record the pressure-time (P-t) curve and thrust-time (F-t) curve. The measurement results are as Figure 1 shown, and the pressure boosting rates of the two groups of tests are as Figure 2 shown.

[0085] (3)截取工作段,将燃烧时间分为34个时间段,分段计算平均压强和总冲,依据冲量法原理对各时间段燃速r进行计算,燃速测试结果如 Figure 3 所示。

[0086] In step (2), this patent is introduced to judge whether the combustion can instantaneously respond to the change in pressure during the test process.

[0087] The burning rate coefficient of composite propellant 1 can be calculated using the Vieri formula based on the experimental results. And the pressure index n, respectively, are obtained by solving for the pressure value and the burning rate value. See the curve graph. Figure 4 As shown. For Linear fitting of the curves shows that when the pressure is between 5.61 and 32.44 MPa, the propellant's burning rate coefficient is 6.868 and the pressure exponent is 0.412; when the pressure is between 32.44 and 74.96 MPa, the propellant's burning rate coefficient is 2.046 and the pressure exponent is 0.761.

[0088] Composite propellant 2 See the curve graph Figure 5 As shown. For Linear fitting of the curves shows that when the pressure is between 5.83 and 9.39 MPa, the propellant's burning rate coefficient is 5.112 and the pressure exponent is 0.57; when the pressure is between 9.39 and 20.18 MPa, the propellant's burning rate coefficient is 7.216 and the pressure exponent is 0.408; and when the pressure is between 20.18 and 32.62 MPa, the propellant's burning rate coefficient is 4.594 and the pressure exponent is 0.558.

[0089] The critical boost rate in the two sets of experiments was calculated according to equation (7). For composite propellant 1, the pressure index n was 0.41 at 30 MPa, the critical boost rate was 146 MPa / s, and the actual boost rate was 161 MPa / s. The actual boost rate was greater than the critical boost rate, indicating that the burning rate lags behind the pressure change. For composite propellant 2, the pressure index n was 0.39 at 30 MPa, the critical boost rate was 154 MPa / s, and the actual boost rate was 73 MPa / s. The boost rate was less than the critical value, indicating that the propellant combustion energy responds instantaneously to the pressure change.

[0090] The test results of both sets of experiments also prove the rationality of the above analysis. For example... Figure 3 As shown, under the same pressure, the measured burning rate of composite propellant 2 is lower than that of composite propellant 1. This indicates that because the pressurization rate of composite propellant 2 is too high, combustion cannot respond instantaneously to pressure changes. The measured pressure is not the burning rate at the current pressure, but the burning rate at a lower pressure. Therefore, the burning rate exhibits a lag and is lower than expected.

Claims

1. A method for determining the boost rate boundary conditions in the impulse-based combustion rate test, characterized in that, Includes the following steps: Step S1: 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; Step S2: 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. By rearranging the expression for the rate of change of burning rate, we obtain the dynamic pressure burning rate and the dynamic relationship between the rate of change of burning rate and pressure: ；（3） Equation (3) is a classical first-order dynamic differential equation. The dynamic relationship between dynamic pressure, combustion rate, and 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 equation (3); Step S3: Determine the boundary conditions for dynamic pressure burning rate and timely response to pressure changes; Step S4: 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. The steps for obtaining the boundary conditions for the boost rate are as follows: Differentiating Vieri's formula with respect to time yields: （5） Substituting equation (5) into the boundary conditions for dynamic pressure combustion rate and timely response to pressure changes, the boundary conditions for the boost rate are obtained as follows: （6） in For boost rate; Step S5: Determine the critical value and range of boost rate.

2. The method for determining the boost rate boundary condition in the impulse method combustion rate test according to claim 1, characterized in that, The steps to obtain the rate of change of burning rate are as follows: The rate of change of combustion speed for: ；（2） in n is the burn rate coefficient; n is the pressure exponent; It characterizes the degree of lag in the response of dynamic pressure burning rate to pressure changes.

3. The method for determining the boost rate boundary condition in the impulse-based combustion rate test according to claim 1, characterized in that, The boundary condition for determining the dynamic pressure burning rate and responding promptly to pressure changes is as follows: （4） When the requirements of equation (4) are met, the dynamic pressure burning rate can respond to the change of pressure 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.

4. The method for determining the boost rate boundary condition in the impulse method combustion rate test according to claim 1, characterized in that, The steps for determining the critical value and range of boost rate are as follows: 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. The critical value for boost rate is: ；（7） in It is 0.005s.

5. 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 method for determining the boost rate boundary conditions in the impulse method combustion rate test according to any one of claims 1-4.

6. A computer-readable storage medium storing a computer program; characterized in that, When the computer program is executed by the processor, it implements the method for determining the boost rate boundary conditions in the impulse method combustion rate test according to any one of claims 1-4.

Images