Design method of nozzle for ultrasonic molecular beam injection system of quasi-annular symmetric stellarator

Abstract

The present application relates to the field of feeding of magnetic confinement nuclear fusion equipment, and provides a design method for the nozzle of the ultrasonic molecular beam injection system of a quasi-annular symmetric stellarator. Based on the physical properties of plasma and the geometric structure of the magnetic confinement fusion device, the volume average density of the plasma in the device under the preset gas source pressure is estimated through the experimental data of the Japanese LHD device, and the gas injection rate of the quasi-annular symmetric stellarator is calculated accordingly. Then, by calculating the radius of the throat of the Laval nozzle and analyzing the state of the gas flow in the nozzle, the Mach number at the nozzle nozzle is obtained to ensure the maximum energy conversion efficiency during the gas state transition. On this basis, the Foelsch method is used to construct an optimized Laval nozzle physical model through the detailed design of the three parts from the nozzle inlet to the throat, the throat to the inflection point, and the inflection point extending to the nozzle outlet. The isentropic expansion efficiency and high-speed flow performance of the gas in the nozzle are improved.

Description

Technical Field

[0001] The present application relates to the technical field of charging of magnetic confinement nuclear fusion equipment, and in particular to a design method for a nozzle of an ultrasonic molecular beam injection system of a quasi-annular symmetric stellarator. Background Art

[0002] The contents of this section merely provide background information related to the present application and may not constitute prior art.

[0003] Magnetic confinement fusion energy is one of the most promising new energy sources to promote the progress of human civilization in today's world. However, the establishment of magnetic confinement fusion reactors still faces a series of major challenges that need to be solved urgently, among which the fuel supply method is a key link. The fuel supply system is not only the core means to achieve plasma particle control, but also the basis for maintaining effective fusion reactions and energy output. The system needs to continuously provide two key fuels, deuterium and tritium, to the fusion plasma to ensure the continuation of the fusion reaction.

[0004] In the existing technology, gas injection and particle injection are two mainstream and widely explored methods. However, each of these two methods has significant limitations. Although the gas injection method has advantages in ease of operation and equipment complexity, it performs poorly in terms of fuel efficiency and it is difficult to achieve the goal of efficient fuel utilization. In contrast, the particle injection method can theoretically achieve a near-perfect 100% fuel efficiency, a feature that makes it extremely attractive in certain high-end applications. However, the shortcomings of the particle injection method are also prominent: the equipment required is extremely complex, and the manufacturing and maintenance costs of the particles are expensive. These factors together constitute a huge obstacle to its economic feasibility and widespread promotion.

[0005] As an innovative fuel supply method, the ultrasonic molecular beam injection system has gradually emerged with its unique advantages. The system cleverly combines the advantages of gas injection and particle injection. Through the design of the Laval nozzle, it achieves a deeper injection depth of the fuel and can directly deliver the fuel to the inside of the plasma. The Laval nozzle accelerates the gas to Mach speed through the principle of isentropic expansion, so that it can quickly reach the edge of the plasma and penetrate to a certain depth in a direction. Compared with the diffusion supply of traditional gas injection, this supply method significantly improves the utilization rate of fuel and demonstrates the huge potential of the ultrasonic molecular beam injection system in fuel supply efficiency.

[0006] However, in the specific application scenario of the Chinese First Quasi-axisymmetric Stellarator (CFQS), the performance of the ultrasonic molecular beam injection system faces new challenges. Due to the particularity of the structure of the quasi-axisymmetric stellarator, the shape of the Laval nozzle has a crucial influence on its gas injection efficiency. Even slight differences in the shape of the nozzle may lead to significant fluctuations in the gas injection efficiency. Unfortunately, the prior art still lacks a method that can accurately calculate the gas injection volume and optimize the nozzle size to design the Laval nozzle that is most suitable for the application of quasi-axisymmetric stellarators. This technical bottleneck seriously restricts the performance and wide application of the ultrasonic molecular beam injection system in quasi-axisymmetric stellarators. Therefore, there is an urgent need for a design method for the nozzle of the ultrasonic molecular beam injection system of the quasi-axisymmetric stellarator, which has become a key issue that needs to be solved. Summary of the invention

[0007] In order to solve the above technical problems, the purpose of this application is to provide a design method for the nozzle of the ultrasonic molecular beam injection system of a quasi-annular symmetric stellarator, which improves the gas flow and transportation efficiency by calculating the gas injection volume and optimizing the nozzle size.

[0008] The purpose of this application is achieved through the following technical solutions:

[0009] In a first aspect, the present invention provides a method for designing a nozzle of a quasi-annular symmetric stellarator ultrasonic molecular beam injection system, comprising:

[0010] The volume average density of the plasma in the quasi-annular symmetric stellarator under the preset gas source pressure is calculated by using a preset formula; the preset formula is obtained by converting the volume average density formula of the plasma in the LHD device based on the electron density distribution shape in the quasi-annular symmetric stellarator;

[0011] By using the first relationship between the gas injection rate and the gas source pressure of the LHD device, the total number of particles in the LHD device and the quasi-annular symmetric stellarator is calculated respectively by knowing the volume and the volume average density of the plasma in the LHD device and the quasi-annular symmetric stellarator under fixed gas pressure injection conditions; by comparing the number of particles with the gas injection rate, the injection rate of the gas in the quasi-annular symmetric stellarator per unit time under the preset gas source pressure is obtained;

[0012] The throat radius of the Laval nozzle is calculated according to the second relationship between the injection rate of the gas in the quasi-annular symmetric stellarator and the throat area of ​​the Laval nozzle;

[0013] An ideal model of the relationship between Mach number and pressure is established based on isentropic flow. Based on the ideal model, the Mach number of the Laval nozzle is calculated when the gas reaches the maximum velocity at the nozzle of the Laval nozzle according to the critical pressure and critical temperature corresponding to the maximum velocity.

[0014] The Laval nozzle physical model is established according to the Mach number and the throat radius. The Laval nozzle physical model is divided into three parts based on the Foelsch method according to the throat position and the inflection point position of the Laval nozzle physical model, among which:

[0015] The first part of the Laval nozzle is the area between the nozzle inlet and the throat, and the cross section of the first part can adopt any curve;

[0016] The second part of the Laval nozzle is the area between the throat and the inflection point. The Prandtl-Meyer function is used to describe the third relationship between the Mach number and the speed at each point on the nozzle boundary curve, and the expansion angle is calculated based on the Mach number at the nozzle outlet and the Mach number at the inflection point. The inflection point coordinates are solved according to the relationship between the nozzle cross section and the Mach number. The cross-sectional curve equation of the second part is established based on the combined relationship between the expansion angle, the inflection point coordinates and the second part.

[0017] The third part of the Laval nozzle is the area between the inflection point and the nozzle outlet. The cross-sectional curve equation of the third part is constructed through the geometric relationship of the third part on the physical model of the Laval nozzle and preset constraints. The preset constraints are: the velocity vectors of the Mach line emitted along the inflection point are collinear; an arc-shaped auxiliary line is made to pass vertically through the cross-sectional curve equation, and the Mach number of the gas at each point on the auxiliary line is equal; on the Mach line with the inflection point as the starting point, the Mach number of any point is related to the radius of the vector center, and the vector center is the center of the auxiliary line.

[0018] Furthermore, the preset formula is constructed by the following steps:

[0019] Based on the LHD device ultrasonic molecular beam injection experiment, the plasma density profile data was measured under the preset gas source pressure, and the plasma density profile formula used to describe the relationship between the plasma density and the average small radius of the LHD device was obtained by fitting.

[0020] The plasma density profile formula is integrated within the volume of the LHD device to obtain the volume average electron density formula of the LHD device;

[0021] Based on the electron density distribution shape in the quasi-annular symmetric stellarator, the preset formula is calculated according to the volume average electron density formula of the LHD device.

[0022] Furthermore, the preset formula is:

[0023]

[0024] in, is the volume average density of the plasma in the quasi-annular symmetric stellarator, is the normalized minor radius, is the normalization coefficient, is the volume of the quasi-annular symmetric stellarator plasma, is the large radius of the quasi-annular symmetric stellarator, is the plasma density at the magnetic axis of the quasi-annular symmetric stellarator, The average small radius is The average plasma density in the quasi-annular symmetric stellarator at .

[0025] Further, the steps of calculating the Mach number when the Laval nozzle conforms to isentropic flow specifically include:

[0026] The range of Knudsen number is determined based on the gas flow conforming to the continuum medium;

[0027] According to the relationship between the range of Knudsen number, the mean free path of gas molecules and the pressure of gas, the Mach number of Laval nozzle under critical pressure when it meets isentropic flow is calculated based on the ideal model.

[0028] Furthermore, the step of calculating the Mach number when the Laval nozzle conforms to isentropic flow also includes:

[0029] The critical temperature is determined based on the triple point of the gas. According to the relationship between the critical temperature, gas source temperature, specific heat ratio of the gas and the Mach number, the Mach number of the Laval nozzle at the critical temperature when it conforms to isentropic flow is calculated.

[0030] Furthermore, the cross-sectional curve equation of the second part is:

[0031]

[0032] in, is the throat radius, is the expansion angle, and are the velocity at the Laval nozzle outlet and the velocity at the inflection point, respectively. is the horizontal coordinate of the inflection point on the curve, are the coordinates of any point on the curve.

[0033] Furthermore, the cross-sectional curve equation of the third part is:

[0034]

[0035] in, It's point P At Mach angle, For point P The Mach number at point P is located on the Mach line at the inflection point. For points P and Q The distance between is the angle at point P, and are the radius vector corresponding to any point in the cross section of the Laval nozzle physical model and the radius vector at the inflection point, and are the cross-sectional area corresponding to any point in the cross section of the Laval nozzle physical model and the cross-sectional area at the inflection point, and are the nozzle exit velocity and the Mach number at a specific cross section in the third section. speed, is the expansion angle.

[0036] In a second aspect, the present invention provides a design system for a nozzle of a quasi-annular symmetric stellarator ultrasonic molecular beam injection system, comprising:

[0037] The volume average density calculation module is used to calculate the volume average density of the plasma in the quasi-annular symmetric stellarator under the preset gas source pressure using a preset formula; the preset formula is obtained by converting the volume average density formula of the plasma in the LHD device based on the electron density distribution shape in the quasi-annular symmetric stellarator;

[0038] A gas injection rate calculation module is used to calculate the total number of particles in the LHD device and the quasi-annular symmetric stellarator respectively by using the first relationship between the gas injection rate and the gas source pressure of the LHD device, by knowing the volume and the average volume density of the plasma in the LHD device and the quasi-annular symmetric stellarator under fixed gas pressure injection conditions; and by comparing the number of particles with the gas injection rate, the injection rate of the gas in the quasi-annular symmetric stellarator per unit time under a preset gas source pressure is obtained;

[0039] A throat radius calculation module, used for calculating the throat radius of the Laval nozzle according to a second relationship between the injection rate of the gas in the quasi-annular symmetric stellarator and the throat area of ​​the Laval nozzle;

[0040] A Mach number calculation module is used to establish an ideal model of the relationship between Mach number and pressure based on isentropic flow, and to calculate the Mach number of the Laval nozzle when it meets isentropic flow based on the critical pressure and critical temperature corresponding to the maximum velocity of the gas at the nozzle of the Laval nozzle based on the ideal model;

[0041] The nozzle cross-section curve calculation module is used to establish a Laval nozzle physical model according to the Mach number and the throat radius. The Laval nozzle physical model is divided into three parts based on the Foelsch method according to the throat position and the inflection point position of the Laval nozzle physical model, among which:

[0042] The first part of the Laval nozzle is the area between the nozzle inlet and the throat, and the cross section of the first part can adopt any curve;

[0043] The second part of the Laval nozzle is the area between the throat and the inflection point. The Prandtl-Meyer function is used to describe the third relationship between the Mach number and the speed at each point on the nozzle boundary curve, and the expansion angle is calculated based on the Mach number at the nozzle outlet and the Mach number at the inflection point. The inflection point coordinates are solved according to the relationship between the nozzle cross section and the Mach number. The cross-sectional curve equation of the second part is established based on the combined relationship between the expansion angle, the inflection point coordinates and the second part.

[0044] The third part of the Laval nozzle is the area between the inflection point and the nozzle outlet. The cross-sectional curve equation of the third part is constructed through the geometric relationship of the third part on the physical model of the Laval nozzle and preset constraints. The preset constraints are: the velocity vectors of the Mach line emitted along the inflection point are collinear; an arc-shaped auxiliary line is made to pass vertically through the cross-sectional curve equation, and the Mach number of the gas at each point on the auxiliary line is equal; on the Mach line with the inflection point as the starting point, the Mach number of any point is related to the radius of the vector center, and the vector center is the center of the auxiliary line.

[0045] In a third aspect, the present invention provides an electronic device comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, the steps corresponding to the method in the first aspect are implemented.

[0046] In a fourth aspect, the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps corresponding to the method in the first aspect.

[0047] In summary, the technical solution of the embodiment of the present application has at least the following advantages and beneficial effects:

[0048] Based on the physical properties of plasma and the geometric structure of the magnetic confinement fusion device, the present invention accurately estimates the volume average density of plasma in the device under the corresponding gas source pressure through the experimental data of the Japanese LHD device, and further calculates the gas injection rate of the quasi-ring-symmetric stellarator based on this. Then, by calculating the throat radius of the Laval nozzle and deeply analyzing the gas flow state in the nozzle, the Mach number at the nozzle nozzle is obtained to ensure the maximum energy conversion during the gas state transition. On this basis, the Foelsch method is used to construct an optimized Laval nozzle physical model through the detailed design of the three parts from the nozzle inlet to the throat, the throat to the inflection point, and the inflection point to the nozzle outlet. The isentropic expansion efficiency and high-speed flow performance of the gas in the nozzle are improved, and the stability and adaptability of the nozzle under different working conditions are enhanced, thereby effectively improving the plasma confinement performance and fuel supply efficiency of the magnetic confinement fusion device, and providing strong technical support for the research and development of magnetic confinement fusion energy. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 A flow chart of a method for designing a nozzle of an ultrasonic molecular beam injection system of a quasi-annular symmetric stellarator provided by the present invention;

[0050] Figure 2 is a graph showing the relationship between the gas flow rate of hydrogen and the gas source pressure in the LHD device of the present invention;

[0051] Figure 3 Schematic diagram of the throat of the Laval nozzle of the present invention;

[0052] Figure 4 It is a schematic diagram of the Laval nozzle under ideal conditions in the present invention;

[0053] Figure 5 It is a cross-sectional schematic diagram of the physical model of the Laval nozzle in the present invention;

[0054] Figure 6 : is a diagram showing the relationship between the axial and radial dimensions of the Laval nozzle at different Mach numbers in the present invention;

[0055] Figure 7 : is a diagram showing the relationship between the axial and radial dimensions of the Laval nozzle at different expansion angles in the present invention;

[0056] Figure 8 The present invention provides a schematic structural diagram of a design system for a nozzle of an ultrasonic molecular beam injection system of a quasi-annular symmetric stellarator. DETAILED DESCRIPTION

[0057] In order to make the purpose, technical solution and advantages of the embodiments of the present application clearer, the technical solution in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are part of the embodiments of the present application, rather than all the embodiments. The components of the embodiments of the present application described and shown in the drawings here can be arranged and designed in various different configurations.

[0058] Please refer to Figure 1 The present invention provides a method for designing a nozzle of an ultrasonic molecular beam injection system of a quasi-axisymmetric stellarator, which is applied to the design of a nozzle of an ultrasonic molecular beam injection system of the first Chinese quasi-axisymmetric stellarator (CFQS), and includes:

[0059] S101, using a preset formula, calculating the volume average density of the plasma in the quasi-annular symmetric stellarator under a preset gas source pressure; the preset formula is obtained by converting the volume average density formula of the plasma in the LHD device based on the electron density distribution shape in the quasi-annular symmetric stellarator.

[0060] It should be noted that the volume-averaged electron density is a parameter used to describe the average level of electron distribution in the entire system or volume of matter. It reflects the overall state of electron distribution in a specific substance to reflect the distribution of electrons. It is obtained based on the physical properties of plasma and the geometric structure of the magnetic confinement fusion device of the quasi-annular symmetric stellarator. In the magnetic confinement fusion device, the plasma is confined by the magnetic field in a specific area, and the shape and size of the area directly determine the volume of the plasma. The average density of plasma, that is, the number of particles per unit volume, is a core parameter in fusion reactions because it has a direct impact on the rate and efficiency of fusion reactions.

[0061] Specifically, the LHD device is a Japanese Large Helical Device (LHD device for short). Based on the LHD device ultrasonic molecular beam injection experiment, the plasma density profile data was measured under a preset gas source pressure, and the plasma density profile formula was obtained by fitting a fifth-order polynomial. This formula is used to describe the relationship between the plasma density and the average small radius of the LHD device. The specific formula is as follows:

[0062]

[0063] in, is the fitting coefficient, , is the normalized minor radius, is the normalization coefficient, Substituting into formula (1) we get:

[0064]

[0065] in is the coefficient,

[0066] The plasma density profile formula (2) is integrated within the volume of the LHD device to obtain the volume average electron density formula of the LHD device; the specific formula is as follows:

[0067]

[0068] in, is the volume average density of plasma in the LHD device, is the volume of plasma in the LHD device, is the maximum radius of the LHD device, The average small radius is Plasma density in the LHD device at .

[0069] When calculating the average density of the plasma in a quasi-annular symmetric stellarator, since the electron density distribution in the quasi-annular symmetric stellarator is approximately a parabolic distribution with a central peak, the electron density profile distribution is approximately: Therefore, according to the electron density distribution shape in the quasi-annular symmetric stellarator, the preset formula is calculated according to the volume average electron density formula (4) of the LHD device. The preset formula is:

[0070]

[0071] in, is the volume average density of the plasma in the quasi-annular symmetric stellarator, is the normalized minor radius, is the normalization coefficient, is the volume of the quasi-annular symmetric stellarator plasma, is the large radius of the quasi-annular symmetric stellarator, is the plasma density at the magnetic axis of the quasi-annular symmetric stellarator, The average small radius is The average plasma density in the quasi-annular symmetric stellarator at .

[0072] S102, using the first relationship between the gas injection rate and the gas source pressure of the LHD device, by knowing the volume of the LHD device and the quasi-annular symmetric stellarator and the volume average density of the plasma under fixed gas pressure injection conditions, calculate the total number of particles in the LHD device and the quasi-annular symmetric stellarator respectively; by comparing the number of particles with the gas injection rate, the injection rate of gas in the quasi-annular symmetric stellarator per unit time under the preset gas source pressure is obtained.

[0073] Specifically, this step aims to estimate the amount of gas injected into the quasi-annular symmetric stellarator per unit time under the corresponding gas source pressure, that is, to estimate the gas injection rate in the quasi-annular symmetric stellarator. In order to achieve this goal, the data obtained from the ultrasonic molecular beam injection experiment in the LHD device are referred to. These data show a direct relationship between the gas injection rate (that is, the gas velocity of hydrogen in the LHD device) and the gas source pressure, and a relationship curve is drawn (such as Figure 2 Through observation and analysis Figure 2 , the gas injection rate corresponding to any given gas pressure can be determined.

[0074] Next, using experimental data obtained under fixed gas pressure injection conditions, including the volume average density of the plasma in the device and the volume of the entire device, the total number of plasma particles inside the device was calculated based on this information.

[0075] Finally, in order to obtain the gas injection rate of the quasi-annular symmetric stellarator, an analogy method is used. Specifically, the supersonic molecular beam gas injection rate in the quasi-annular symmetric stellarator is compared with the supersonic molecular beam injection rate in the LHD device. Through this comparison process, the gas injection rate of the quasi-annular symmetric stellarator is inferred, thus completing the entire estimation process. The specific formula is as follows:

[0076]

[0077] in, is the gas injection rate of the LHD device, is the average density of plasma in the LHD device, is the average density of plasma in the quasi-annular symmetric stellarator, is the total number of particles in the quasi-annular symmetric stellarator plasma, is the total number of plasma particles in the Japanese LHD device, is the volume of plasma in the Japanese LHD device, is the volume of the plasma in the quasi-annularly symmetric stellarator.

[0078] S103, calculating the throat radius of the Laval nozzle according to the second relationship between the injection rate of the gas in the quasi-annular symmetric stellarator and the throat area of ​​the Laval nozzle.

[0079] Specifically, in the process of gas flow, a key transition point occurs in the throat area. Specifically, when the gas flows through the throat and reaches its minimum radius, the airflow that was originally at subsonic speed will change significantly and turn into supersonic flow. Conversely, if the gas is already a supersonic flow, when it passes through the throat, it will slow down and turn into a subsonic flow. This phenomenon shows that the throat plays a vital role in the process of accelerating the gas to supersonic speed and is the only way to achieve supersonic flow. Based on this principle, in the throat of the nozzle, the flow state of the airflow is very special, and its Mach number M is exactly equal to 1, which means that the speed at this time is equal to the speed of sound, and the airflow propagates in the throat in the form of sound waves. Therefore, it can be known that the throat is not only a key node for the change of gas velocity, but also the core of the realization of supersonic flow. The area of ​​the throat is ,like Figure 3 As shown, the design is a circular cross section. Using the quasi-annular symmetric stellarator gas injection rate obtained in step S102 , throat area of ​​Laval nozzle The relationship between the gas injection rate and the throat nozzle area can be obtained:

[0080]

[0081] in, is the gas injection rate, is the gas source pressure (in Pa). represents the nozzle area, such as Figure 3 Shown (in m²). is the source gas temperature (in K), is the specific heat ratio of the gas (1.4 for hydrogen), and is the specific gas constant, which is 4124 (J / kg·K) for hydrogen. is the throat area of ​​the Laval nozzle, is the throat radius of the Laval nozzle.

[0082] S104, based on the isentropic flow, an ideal model of the relationship between the Mach number and the pressure is established, and based on the ideal model, the Mach number of the Laval nozzle when it meets the isentropic flow is calculated using the critical pressure and critical temperature corresponding to the maximum velocity of the gas at the nozzle of the Laval nozzle.

[0083] Specifically, under given conditions, the ideal Mach number can be obtained through the relationship between the Mach number and pressure. Based on the assumption of isentropic flow, the relationship between the Mach number and pressure (i.e., the ideal model) is:

[0084]

[0085] in, is the gas source pressure, is the pressure at the nozzle, is the Mach number, is the specific heat ratio of the gas.

[0086] Due to the background pressure in the vacuum during gas injection Usually in 10 -3 Pa level, under ideal conditions, the pressure at the gas injection nozzle should match the background pressure, such as Figure 4 As shown, the Mach number at the Laval nozzle under ideal conditions can be obtained as follows:

[0087]

[0088] In the actual environment, the state of the gas is not fixed, but will change with the changes in pressure and temperature. This change often means that the gas no longer meets the conditions of isentropic flow, that is, the ideal state in which the entropy remains constant during the flow of the gas no longer holds. When the state of the gas changes, its originally stored energy is no longer completely converted into the kinetic energy of the molecules, but a part of it is used for the change of the state of the gas, such as the expansion or compression of the volume, the rise and fall of the temperature, etc. In order to maximize the speed of the gas at the nozzle, it is necessary to ensure that the state of the gas at the nozzle is at the critical point of the transition. These critical points mark the moment when the gas state changes most dramatically, and are also the points where the gas energy conversion efficiency is the highest. When the conditions of the nozzle precisely reach these critical points, the potential energy that can be released by the gas during the flow process will be converted into the kinetic energy of the molecules to the greatest extent, thereby maximizing the gas speed.

[0089] Specifically, during the gas transformation process, the Mach number calculation process corresponding to the critical pressure is as follows:

[0090] The range of Knudsen number is determined based on the gas flow conforming to the continuum medium. Less than 0.01, according to the relationship between the range of Knudsen number, the mean free path of gas molecules and the pressure of gas, the Mach number of Laval nozzle under critical pressure when it meets isentropic flow is calculated based on the ideal model. The critical pressure is obtained according to the known formula, which is as follows:

[0091]

[0092] in, is the Knudsen number, which describes the ratio of the mean free path of gas molecules to the nozzle diameter; is the mean free path of gas molecules; is the nozzle diameter; is the Boltzmann constant; is the temperature in K; is the molecular collision cross section, in m; is the critical pressure at the nozzle, in Pa;

[0093] Therefore, the critical pressure at the nozzle With air source pressure The ratio formula is:

[0094]

[0095] According to formula (12), we can calculate The value is 100Pa, and the gas source pressure is 0.5MPa, the air source velocity is 0, and the air source temperature is The temperature is 300K, the specific heat ratio of the gas is 1.4, and the above parameters are substituted into formula (13) to calculate the Mach number to be approximately 7.

[0096] In the process of gas transformation, the Mach number corresponding to the critical temperature is determined by the triple point of the gas. According to the relationship between the critical temperature, gas source temperature, specific heat ratio of the gas and the Mach number, the Mach number of the Laval nozzle at the critical temperature when it meets the isentropic flow is calculated. Specifically:

[0097]

[0098] in, is the critical temperature at the nozzle, which is 13.8K. Substituting it into formula (14), the Mach number is approximately 10. It can be seen that according to the calculation of the critical condition, the value range of the Mach number is between 7 and 10.

[0099] S105, a Laval nozzle physical model is established according to the Mach number and the throat radius. The Laval nozzle physical model is divided into three parts according to the throat position and the inflection point position of the Laval nozzle physical model based on the Foelsch method, such as Figure 5 As shown, the cross-sectional curve equations are established according to the three parts.

[0100] Specifically, using the Foelsch method, we only need to determine the radius of the throat of the Laval nozzle: , Mach number M at the nozzle, specific heat ratio of gas γ and inflection point The expansion angle By substituting these parameters into the cross-sectional curve equation, the boundary model can be obtained.

[0101] The first part of the Laval nozzle is the area between the nozzle inlet and the throat, i.e. Figure 5 The leftmost to =1 (i.e. ), which can also be regarded as the distance from the leftmost end of the nozzle outer wall to point In the region of FIG. 1 , the cross section of the first portion may adopt an arbitrary curve.

[0102] The second part of the Laval nozzle is the area between the throat and the inflection point, called the expansion section, i.e. Figure 5 middle =1 (i.e. ) position to the inflection point, where an expansion wave is generated and propagates downstream through reflection from the relative wall. Therefore, the Prandtl-Meyer function is used to describe the third relationship between the Mach number and the speed at each point on the nozzle boundary curve, and the expansion angle is calculated based on the Mach number at the nozzle outlet and the Mach number at the inflection point; the inflection point coordinates are solved based on the relationship between the nozzle cross section and the Mach number; based on the combined relationship between the expansion angle, the inflection point coordinates and the second part, the cross-sectional curve equation of the second part is established.

[0103] Specifically, the design of the expansion section requires first solving the expansion angle According to the Prandtl-Meyer function, the relationship between the Mach number and the speed at each point on the Laval nozzle boundary curve can be obtained as follows:

[0104]

[0105] in, is the Mach number, is the specific heat ratio of the gas, is the velocity of each point on the Laval nozzle boundary curve. Substituting the Mach number at the nozzle outlet and the Mach number at the inflection point into the above equation, we can get the velocity at the nozzle outlet: and the speed at the turning point , and then according to Find the expansion angle .

[0106] Secondly, solve the inflection point coordinates The Mach number at any location in the duct is a function of the ratio of the local duct area to the sonic throat area. According to the area-Mach number relationship, the cross-sectional area of ​​the inflection point nozzle is and throat area The relationship between them is as follows:

[0107]

[0108] in is the Mach number, is the specific heat ratio of the gas, is the ordinate of the inflection point, is the throat radius, is the expansion angle. According to the cross-sectional area of ​​the nozzle at the inflection point and throat area The relationship between the vertical coordinates of the inflection point is obtained Finally, substitute it into the curve equation to get the horizontal coordinate of the inflection point , thus finding the cross-sectional curve equation of the second part, as follows:

[0109]

[0110] in, is the throat radius, is the expansion angle, and are the velocity at the Laval nozzle outlet and the velocity at the inflection point, respectively. is the horizontal coordinate of the inflection point on the curve, are the coordinates of any point on the curve.

[0111] The third part of the Laval nozzle is the area between the inflection point and the nozzle outlet, which is called the straightening section. Figure 5 In the area to the right of the inflection point, all the expansion waves generated by the expansion section in the straightening section are offset. Therefore, the cross-sectional curve equation of the third part is constructed through the geometric relationship of the third part on the Laval nozzle physical model and the preset constraints. The preset constraints are: the velocity vectors of the Mach line emitted along the inflection point are collinear; an arc-shaped auxiliary line H is made to pass vertically through the cross-sectional curve equation, and the Mach number of the gas at each point on the auxiliary line is equal (that is, M=M1), and the point where the auxiliary line H passes through the x-axis is ; On the Mach line ML with the inflection point as the starting point, the Mach number of any point is related to the radius of the vector center, and the vector center is the center of the auxiliary line.

[0112] Specifically, in the constraint condition, the velocity vectors along the Mach lines emitted from the inflection point are collinear. The Mach line is a virtual line used to describe the boundary between the supersonic region and the subsonic region in the fluid, which represents the position where the fluid velocity reaches the local speed of sound. In the straightening section of the Laval nozzle, all the Mach line velocity vectors emitted from the inflection point remain collinear, which means that in this area, the direction and velocity changes of the gas flow must be consistent to avoid unnecessary vortices or velocity gradients, thereby ensuring the continuity and stability of the gas flow. This constraint condition helps to optimize the flow field structure of the nozzle and improve the jet efficiency.

[0113] Draw an arc-shaped auxiliary line perpendicular to the cross-sectional curve equation, and the Mach number of the gas at each location on the auxiliary line is equal. The purpose is to introduce an arc-shaped auxiliary line as an analysis tool, which is perpendicular to the cross-sectional curve and passes through the inflection point. The Mach number is the ratio of the fluid velocity to the local sound speed, and is an important parameter for measuring the flow state of the fluid. Under this condition, the Mach number of the gas is required to be the same at any position on the auxiliary line, which means that the flow state (especially the velocity) of the gas is uniform on the specific path represented by the auxiliary line. This assumption helps to simplify the problem analysis, so that certain characteristics of the cross-sectional curve can be directly derived through geometric relationships, while ensuring the symmetry of the flow field inside the nozzle, which is conducive to reducing energy loss.

[0114] The Mach number of any point on the Mach line ML starting from the inflection point is related to the radius of the vector center, and the vector center is the center of the auxiliary line H, such as Figure 5 As shown. It aims to directly link the Mach number with the nozzle geometry and emphasizes the influence of geometric parameters in nozzle design on the gas flow characteristics. In the Laval nozzle, the inflection point is not only the turning point of the flow channel shape change, but also the key point where the gas flow state changes significantly. By stipulating the relationship between the Mach number on the Mach line starting from the inflection point and the center radius of the auxiliary line (i.e., the vector circle), the acceleration process of the gas in the nozzle can be accurately understood.

[0115] Based on the above preset constraints, in the Laval nozzle physical model (such as Figure 5 As shown in Figure 2, the geometric relationship of each point is analyzed to establish the cross-sectional curve equation of the third part, which is:

[0116]

[0117]

[0118] in, It's point P At Mach angle, For point P The Mach number at the nozzle is , point P is located on the Mach line at the inflection point, For points P and Q The distance between is the angle at point P, and are the radius vector corresponding to any point in the cross section of the Laval nozzle physical model and the radius vector at the inflection point, and are the cross-sectional area corresponding to any point in the cross section of the Laval nozzle physical model and the cross-sectional area at the inflection point, and are the nozzle exit velocity and the Mach number at a specific cross section in the third section. speed, is the expansion angle.

[0119] Considering the manufacturing accuracy and keeping the nozzle size small to avoid reducing the effective range of the supersonic molecular beam, the throat diameter is fixed to 0.3 mm in the present invention. Set to 1.4. According to the calculation of the critical condition in step S104 above, the Mach number is selected to be between 7 and 10. Since the inflection point angle will affect the convergence of the molecular beam, it is usually preferably set between 3° and 15°. The following tests are conducted on the physical design model of the Laval nozzle corresponding to different Mach numbers M when the inflection point angle is fixed at 5°, and the physical design model of the Laval nozzle corresponding to different expansion angles when the Mach number M is fixed at 10.

[0120] Test 1: Fix the inflection point angle at 5° and then scan different Mach numbers M, such as Figure 6 As shown. It can be seen that with the increase of the design Mach number, the width and length of the nozzle also increase significantly to achieve supersonic flow.

[0121] Experiment 2: Fix the Mach number M to 10 and then scan different expansion angles ,like Figure 7 As shown. It can be seen that, contrary to the increase in Mach number, the increase in the expansion angle at the inflection point will reduce the length of the nozzle and increase the width, which can also be deduced from the relevant formula. However, in practical applications, a smaller expansion angle is usually preferred because a smaller angle can better focus the injected beam, increase its effective range, and be more conducive to entering the plasma.

[0122] Based on the same inventive concept, the present invention provides a design system for a nozzle of a quasi-annular symmetric stellarator ultrasonic molecular beam injection system, comprising:

[0123] The volume average density calculation module 201 is used to calculate the volume average density of the plasma in the quasi-annular symmetric stellarator under a preset gas source pressure using a preset formula; the preset formula is obtained by converting the volume average density formula of the plasma in the LHD device based on the electron density distribution shape in the quasi-annular symmetric stellarator;

[0124] The gas injection rate calculation module 202 is used to calculate the total number of particles in the LHD device and the quasi-annular symmetric stellarator respectively by using the first relationship between the gas injection rate and the gas source pressure of the LHD device, by knowing the volume and the average volume density of the plasma in the LHD device and the quasi-annular symmetric stellarator under fixed gas pressure injection conditions; and by comparing the number of particles with the gas injection rate, the injection rate of the gas in the quasi-annular symmetric stellarator per unit time under a preset gas source pressure is obtained;

[0125] A throat radius calculation module 203, for calculating the throat radius of the Laval nozzle according to a second relationship between the injection rate of the gas in the quasi-annular symmetric stellarator and the throat area of ​​the Laval nozzle;

[0126] The Mach number calculation module 204 is used to establish an ideal model of the relationship between the Mach number and the pressure based on the isentropic flow, and calculate the Mach number of the Laval nozzle when the Laval nozzle meets the isentropic flow based on the critical pressure and critical temperature corresponding to the maximum velocity of the gas at the nozzle of the Laval nozzle.

[0127] The nozzle cross-section curve calculation module 205 is used to establish a Laval nozzle physical model according to the Mach number and the throat radius. The Laval nozzle physical model is divided into three parts based on the Foelsch method according to the throat position and the inflection point position of the Laval nozzle physical model, wherein:

[0128] The first part of the Laval nozzle is the area between the nozzle inlet and the throat, and the cross section of the first part can adopt any curve;

[0129] The second part of the Laval nozzle is the area between the throat and the inflection point. The Prandtl-Meyer function is used to describe the third relationship between the Mach number and the speed at each point on the nozzle boundary curve, and the expansion angle is calculated based on the Mach number at the nozzle outlet and the Mach number at the inflection point. The inflection point coordinates are solved according to the relationship between the nozzle cross section and the Mach number. The cross-sectional curve equation of the second part is established based on the combined relationship between the expansion angle, the inflection point coordinates and the second part.

[0130] The third part of the Laval nozzle is the area between the inflection point and the nozzle outlet. The cross-sectional curve equation of the third part is constructed through the geometric relationship of the third part on the physical model of the Laval nozzle and preset constraints. The preset constraints are: the velocity vectors of the Mach line emitted along the inflection point are collinear; an arc-shaped auxiliary line is made to pass vertically through the cross-sectional curve equation, and the Mach number of the gas at each point on the auxiliary line is equal; on the Mach line with the inflection point as the starting point, the Mach number of any point is related to the radius of the vector center, and the vector center is the center of the auxiliary line.

[0131] Based on the same inventive concept, the present invention provides an electronic device, including: a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, the method for designing the nozzle of the ultrasonic molecular beam injection system of the quasi-annular symmetric stellarator is implemented.

[0132] Based on the same inventive concept, the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the method for designing a nozzle of the ultrasonic molecular beam injection system of the quasi-annular symmetric stellarator.

[0133] The above are only preferred embodiments of the present application and are not intended to limit the present application. For those skilled in the art, the present application may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A method for designing a nozzle of a quasi-annular symmetric stellarator ultrasonic molecular beam injection system, characterized in that: include: Using a preset formula, the volume average density of the plasma in the quasi-annular symmetric stellarator under a preset gas source pressure is calculated; The preset formula is obtained by converting the volume average density formula of the plasma in the LHD device based on the electron density distribution shape in the quasi-annular symmetric stellarator; Utilizing the first relationship between the gas injection rate and the gas source pressure of the LHD device, by knowing the volume of the LHD device and the quasi-annular symmetric stellarator and the volume average density of the plasma under fixed gas pressure injection conditions, the total number of particles in the LHD device and the quasi-annular symmetric stellarator are calculated respectively; by comparing the number of particles with the gas injection rate, the injection rate of the gas in the quasi-annular symmetric stellarator per unit time under a preset gas source pressure is obtained; The throat radius of the Laval nozzle is calculated according to the second relationship between the injection rate of the gas in the quasi-annular symmetric stellarator and the throat area of ​​the Laval nozzle; An ideal model of the relationship between Mach number and pressure is established based on isentropic flow, and based on the ideal model, the Mach number of the Laval nozzle when it meets the isentropic flow is calculated using the critical pressure and critical temperature corresponding to the maximum velocity of the gas at the nozzle of the Laval nozzle; A Laval nozzle physical model is established according to the Mach number and the throat radius. The Laval nozzle physical model is divided into three parts according to the throat position and the inflection point position of the Laval nozzle physical model based on the Foelsch method, wherein: The first portion of the Laval nozzle is the area between the nozzle inlet and the throat, and the cross section of the first portion can adopt any curve; The second part of the Laval nozzle is the area between the throat and the inflection point, the Prandtl-Meyer function is used to describe the third relationship between the Mach number and the speed at each point on the nozzle boundary curve, and the expansion angle is calculated based on the Mach number at the nozzle outlet and the Mach number at the inflection point; the inflection point coordinates are solved according to the relationship between the nozzle cross section and the Mach number; based on the combined relationship between the expansion angle, the inflection point coordinates and the second part, the cross-sectional curve equation of the second part is established; The third part of the Laval nozzle is the area between the inflection point and the nozzle outlet. The cross-sectional curve equation of the third part is constructed through the geometric relationship of the third part on the physical model of the Laval nozzle and preset constraints. The preset constraints are: the velocity vectors of the Mach line emitted along the inflection point are collinear; an arc-shaped auxiliary line is made to vertically pass through the cross-sectional curve equation, and the Mach number of the gas at each point on the auxiliary line is equal; on the Mach line with the inflection point as the starting point, the Mach number of any point is related to the radius of the vector center, and the vector center is the center of the auxiliary line.

2. The method for designing a nozzle of a quasi-annular symmetric stellarator ultrasonic molecular beam injection system according to claim 1, characterized in that: The preset formula is constructed by the following steps: Based on the LHD device ultrasonic molecular beam injection experiment, the plasma density profile data is measured at a preset gas source pressure, and the plasma density profile formula for describing the relationship between the plasma density and the average small radius of the LHD device is obtained by fitting; Integrating the plasma density profile formula within the volume of the LHD device to obtain a volume average electron density formula for the LHD device; Based on the electron density distribution shape in the quasi-annular symmetric stellarator, the preset formula is calculated according to the volume average electron density formula of the LHD device.

3. The method for designing a nozzle of a quasi-annular symmetric stellarator ultrasonic molecular beam injection system according to claim 2, characterized in that: The preset formula is: in, is the volume average density of the plasma in the quasi-annular symmetric stellarator, is the normalized minor radius, is the normalization coefficient, is the volume of the quasi-annular symmetric stellarator plasma, is the large radius of the quasi-annular symmetric stellarator, is the plasma density at the magnetic axis of the quasi-annular symmetric stellarator, The average small radius is The average plasma density in the quasi-annular symmetric stellarator at .

4. The method for designing a nozzle of a quasi-annular symmetric stellarator ultrasonic molecular beam injection system according to claim 1, characterized in that: The step of calculating the Mach number of the Laval nozzle when it complies with isentropic flow specifically includes: The range of Knudsen number is determined based on the gas flow conforming to the continuum medium; According to the relationship formula between the range of the Knudsen number, the mean free path of gas molecules and the pressure of the gas, the Mach number of the Laval nozzle when it meets the isentropic flow at the critical pressure is calculated based on the ideal model.

5. The method for designing a nozzle of a quasi-annular symmetric stellarator ultrasonic molecular beam injection system according to claim 4, characterized in that: The step of calculating the Mach number of the Laval nozzle when it complies with isentropic flow also includes: The critical temperature is determined based on the triple point of the gas, and the Mach number of the Laval nozzle when it complies with isentropic flow at the critical temperature is calculated according to the relationship between the critical temperature, the gas source temperature, the specific heat ratio of the gas and the Mach number.

6. The method for designing a nozzle of a quasi-annular symmetric stellarator ultrasonic molecular beam injection system according to claim 1, characterized in that: The cross-sectional curve equation of the second part is: in, is the throat radius, is the expansion angle, and are the velocity at the Laval nozzle outlet and the velocity at the inflection point, respectively. is the horizontal coordinate of the inflection point on the curve, are the coordinates of any point on the curve.

7. The method for designing a nozzle of a quasi-annular symmetric stellarator ultrasonic molecular beam injection system according to claim 1, characterized in that: The cross-sectional curve equation of the third part is: in, It's point P At Mach angle, Point P The Mach number at point P is located on the Mach line at the inflection point. For points P and Q The distance between is the angle at point P, and are the radius vector corresponding to any point in the cross section of the Laval nozzle physical model and the radius vector at the inflection point, and are the cross-sectional area corresponding to any point in the cross section of the Laval nozzle physical model and the cross-sectional area at the inflection point, and are the nozzle exit velocity and the Mach number at a specific cross section in the third section. speed, is the expansion angle, is the horizontal coordinate of the inflection point on the curve.

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