A method for solving the along-stream parameters of one-dimensional constant cross-section adiabatic friction pipe flow

Abstract

The application relates to a one-dimensional isometric adiabatic friction pipe flow parameter solving method. The method comprises the following steps: collecting the total pressure, total temperature, outlet back pressure and gas mass flow of a pipeline inlet by using a non-intrusive system; iteratively solving the inlet Mach number by using a compressible flow control equation, and calculating the inlet static pressure and static temperature; comparing the outlet critical pressure with the back pressure to judge the flow jam state and determine the outlet boundary condition; combining the boundary condition to solve the outlet Mach number, total pressure and static temperature; solving the total average friction coefficient according to the limit pipe length theorem; setting a step length, taking the inlet Mach number as an initial condition, iteratively solving the along-path Reynolds number by using a fourth-order Runge-Kutta method and correcting the along-path Reynolds number to obtain the along-path distribution of the Mach number; iteratively solving the along-path distribution of the static pressure and velocity based on a gas dynamics method; comparing the predicted value with the measured value of the outlet static pressure, and outputting the along-path curve and the average friction coefficient. The method can obtain the along-path distribution of the one-dimensional isometric adiabatic friction pipe flow parameter.

Description

Technical Field

[0001] This application relates to the field of experimental and numerical solutions in compressible gas dynamics, and in particular to a method for solving friction parameters of one-dimensional adiabatic friction pipe flow with uniform cross-section. Background Technology

[0002] In high-speed aircraft thermal protection systems, the gaseous working fluid used for active cooling is transported through microscale channels within the aircraft. Its mass transport efficiency and outlet flow parameters jointly determine the thermal protection performance, thus affecting the aircraft's safety and reliability. Therefore, accurately obtaining the evolution of the cooling fluid's flow parameters (including density, pressure, and Mach number) within the microchannels is a fundamental prerequisite for evaluating and optimizing the thermal protection system's performance. This flow process can be simplified to a one-dimensional, uniform-section adiabatic friction pipe flow model. During different stages of flight, the inlet and back pressures of the cooling fluid delivery channel continuously change, presenting a series of discrete steady-state conditions. The internal flow state under each condition can be considered a definite, steady distribution. Accurately obtaining the flow parameter distributions under different conditions is crucial for predicting the cooling fluid's transport capacity, calculating outlet flow parameters, and guiding microchannel structure optimization. However, in practical engineering applications, cooling medium delivery channels are generally characterized by their small structural scale, compact spatial layout, and complete enclosure within the aircraft skin. Traditional measurement methods based on invasive sensors are impractical due to the risk of compromising structural integrity and interfering with the flow field, making it difficult to directly obtain flow parameters. This makes it difficult to accurately assess the rationality of the orifice layout during the design phase and to diagnose cooling performance under specific operating conditions during the operational phase, becoming a key technical bottleneck restricting the performance improvement of microchannel flow control systems. Therefore, developing a measurement and solution method capable of non-invasively obtaining the flow parameter distribution along a one-dimensional, uniform cross-section adiabatic friction pipe is of great engineering significance for overcoming the design and verification barriers of microflow control systems and improving the reliability of high-speed aircraft thermal protection systems.

[0003] However, a direct measurement method based on internal multi-point measurements has been disclosed. This method aims to acquire flow path data directly and compare it with theoretical curves. The system includes a gas supply system, an experimental pipe section, and multiple measurement units. These measurement units are arranged at fixed intervals along the pipe flow direction. Correspondingly, multiple measurement holes are opened on the surface of the pipe section, and multiple discrete static pressure sensors, thermocouples, or total pressure probes are installed at each measurement hole location to form a measurement array. During the experiment, after establishing a stable flow by adjusting the gas source and outlet valve, pressure and temperature data from each measurement unit section are simultaneously collected. A discrete distribution map of the flow parameters is plotted, and then a continuous parameter variation curve along the flow path is obtained by fitting the curve. Besides the direct measurement method, there is also a numerical analysis method based on computational fluid dynamics software simulation. This method aims to obtain the complete data distribution of the flow field through numerical calculation. The process includes establishing a computational domain model and configuring a numerical solution system: the computational domain is established based on the three-dimensional geometry of the target flow channel and discretized into a computational grid composed of a large number of elements; the corresponding governing equations, turbulence models, and boundary conditions (inlet total pressure, total temperature, outlet back pressure, adiabatic wall conditions, etc.) are set and loaded into the solution system. After convergence through iterative calculation, continuous distribution cloud maps of flow parameters such as pressure, velocity, and temperature at any location within the entire computational domain can be obtained through post-processing, and parameter data from specific cross-sections or streamlines can be extracted to generate parameter variation curves along the flow path.

[0004] Existing experimental techniques, such as internal multi-point measurement schemes, cannot achieve the ideal premise of "one-dimensional, uniform cross-section, and subject to only uniform friction," resulting in large measurement errors and failing to achieve high-precision, high-resolution measurement of flow parameter distribution along the flow path. Current simulation techniques deviate from the real flow environment, introducing additional errors, and the workload for parameter adjustment and post-processing is substantial and heavily reliant on the operator's experience and expertise. Currently, there is no research scheme for obtaining the parameter distribution along the flow path of one-dimensional, uniform cross-section adiabatic friction pipe flow. Summary of the Invention

[0005] Therefore, it is necessary to provide a method for solving the friction parameters of a one-dimensional adiabatic friction pipe flow that can obtain the friction parameter distribution along the flow path of a one-dimensional adiabatic friction pipe flow with a constant cross-section, in order to address the above-mentioned technical problems.

[0006] A method for solving friction parameters of one-dimensional adiabatic friction pipe flow with uniform cross-section, the method comprising: Step 1: Obtain the geometric parameters of the pipe to be tested and the thermophysical properties of the working fluid inside the pipe. The geometric parameters include cross-sectional area, length, and hydraulic diameter. The thermophysical properties include gas constant and specific heat ratio. Step 2: Collect the total pressure at the pipe inlet, the total temperature at the pipe inlet, the back pressure at the pipe outlet, and the mass flow rate of the gas flowing through the pipe using a non-invasive experimental measurement system; Step 3: Using geometric parameters, thermophysical property parameters, total pressure at pipe inlet, total temperature at pipe inlet, and gas mass flow rate as inputs, simultaneously establish the control equations related to the compressible flow mass flow rate. Solve the control equations using numerical iteration to obtain the Mach number at pipe inlet. Then, calculate the static pressure and static temperature at pipe inlet based on the isentropic relationship. Step 4: Calculate the critical outlet pressure when the outlet Mach number is 1 based on the inlet Mach number and the inlet static pressure. Compare the critical outlet pressure with the pipeline outlet back pressure to determine the congestion state of the flow in the pipeline and determine the corresponding outlet boundary conditions. Step 5: Based on the outlet boundary conditions, the ratio of inlet / outlet static pressure and the isentropic relationship, the flow parameters at the outlet of the pipeline are obtained. The flow parameters at the outlet of the pipeline include the outlet Mach number, the outlet total pressure and the outlet static temperature. Step 6: Based on the limiting pipe length theorem, the overall average friction coefficient is obtained by solving for the inlet Mach number, outlet Mach number, pipe length, and hydraulic diameter. Step 7: Set the calculation step size, take the inlet Mach number as the initial condition for numerical integration, combine the outlet Mach number and the average friction coefficient, and use the fourth-order Runge-Kutta method to iteratively solve the Reynolds number corresponding to each calculation step size. Then, make correction calculations based on the friction coefficient calculation formula until the iteration conditions are met and the Mach number distribution along the process is obtained. Step 8: Based on the gas dynamics control volume method, iteratively solve the flow distribution curves of static pressure, velocity, static temperature and density based on the known Mach number flow distribution; Step 9: Compare and verify the predicted static pressure at the pipeline outlet with the back pressure at the outlet measured in Step 2, and output the inlet section parameters, friction distribution curve, average friction coefficient, and verification error.

[0007] The aforementioned method for solving flow parameters along a one-dimensional, uniform cross-section adiabatic friction pipe utilizes non-invasive boundary measurements combined with multi-step theoretical numerical solutions. It requires only external measurements at the pipe inlet and outlet, eliminating the need for drilling any measurement holes in the pipe's inner wall. This fundamentally eliminates disturbances to the flow field and systematic errors in the measurement results caused by sensor installation. Furthermore, based on compressible aerodynamics theory and high-precision numerical integration methods, it achieves accurate reconstruction of the flow parameter distribution along the pipe within a closed microchannel. This solves the technical problems of large errors, complex operation, and inability to adapt to microscale closed channels inherent in traditional invasive measurements and CFD simulations. Based on limited boundary measurements, a complete continuous flow parameter distribution curve is derived from a high-precision physical model, overcoming the limitation of internal multi-point measurement schemes that can only obtain discrete point data. This provides reliable data support for the design and performance evaluation of microchannels in high-speed aircraft thermal protection systems. Moreover, it eliminates the need for complex sensor arrays, numerous pressure-tapping pipelines, and corresponding acquisition channels, significantly reducing system complexity, construction and maintenance costs, and improving long-term operational reliability. Attached Figure Description

[0008] Figure 1 This is a flowchart illustrating a method for solving the friction parameters of a one-dimensional constant cross-section adiabatic friction pipe flow in one embodiment. Figure 2 This is a schematic diagram of a non-invasive experimental measurement system in one embodiment; Figure 3 This is a flowchart illustrating the solution process in one embodiment; Figure 4 This is a schematic diagram of the limiting tube length theory in one embodiment. Detailed Implementation

[0009] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0010] In one embodiment, such as Figure 1 As shown, a method for solving the friction parameters of one-dimensional adiabatic friction pipe flow with uniform cross-section is provided, including the following steps: Step 1: Obtain the geometric parameters of the pipe to be tested and the thermophysical properties of the working fluid inside the pipe. The geometric parameters include cross-sectional area, length, and hydraulic diameter. The thermophysical properties include gas constant and specific heat ratio.

[0011] Pipeline geometric parameters must be obtained through precise measurements and are known constants. Their values ​​must meet the basic requirements of the one-dimensional flow assumption. The pipe length-to-diameter ratio must be within a specific range to ensure sufficient flow development and that wall friction effects dominate parameter variations along the pipe. Simultaneously, the pipe hydraulic diameter should satisfy the continuous medium assumption; for gaseous working fluids, this diameter typically needs to be greater than the mean free path of molecules. The thermophysical properties of the working fluid can be obtained by consulting relevant property handbooks for the selected gas type. The gas constant is a characteristic parameter of the gas, and the specific heat ratio, the ratio of isobaric specific heat to isochoric specific heat, is a dimensionless parameter.

[0012] Step 2: Collect the total pressure at the pipe inlet, the total temperature at the pipe inlet, the back pressure at the pipe outlet, and the mass flow rate of the gas flowing through the pipe using a non-invasive experimental measurement system.

[0013] like Figure 2As shown, the non-invasive experimental measurement system consists of five parts: a gas source system, a parameter measurement system, an experimental pipe section, a downstream back pressure control system, and a data acquisition and processing system. The entire system only performs external measurements at the pipe inlet and outlet, without the need to drill any measurement holes in the inner wall of the pipe, thus not damaging the integrity of the pipe structure or interfering with the flow field inside the pipe. The gas supply system includes a compressed air source 1, a pressure reducer 2, and a gas dryer 3. A high-pressure gas cylinder provides compressed air. The pressure reducer 2 regulates the stagnation pressure in the upstream chamber. The gas dryer 3 filters and dries the working gas to prevent damage to the flow meter and blockage of the micro-scale pipeline by tiny particles. The parameter measurement system includes a thermal mass flow meter 4 with an accuracy of no less than ±0.5% of full scale, an upstream chamber 5, a pressure sensor 6, and a thermocouple 7. The thermal mass flow meter 4 is installed upstream of the upstream chamber 5, and the measured value is the mass flow rate of the gas flowing through the pipeline. The volume of the upstream chamber 5 is much larger than the gas leakage in the pipeline, ensuring that pressure fluctuations are negligible. Its built-in pressure sensor 6 and thermocouple 7 measure the upstream stagnation pressure and stagnation temperature, respectively. Under pressure conditions below 1 MPa, the stagnation pressure and stagnation temperature are equal to the total pressure and total temperature at the pipeline inlet, respectively. The experimental pipe section includes a solenoid valve 8 and a pipeline model 11. The solenoid valve 8 has a matched diameter. With rapid response and good sealing, it is used to control the gas path opening and closing and to assist in airtightness checks. During the experiment, the parameter measurement system is turned on first and then the solenoid valve 8 is opened. At the end, the data acquisition is stopped first and then the solenoid valve 8 is closed to avoid pressure surges that could damage the sensor. The pipe model 11 is an experimental pipe section that meets the one-dimensional uniform cross-section adiabatic flow conditions. The downstream back pressure control system includes a downstream chamber 12, a pressure controller 13, and a vacuum pump 14. The downstream chamber 12 can be evacuated to a given vacuum level or connected to the atmosphere by the vacuum pump 14 to simulate different back pressure environments from low pressure to normal pressure. The built-in pressure sensor measures the outlet back pressure in real time, and the pressure controller 13 precisely adjusts the downstream chamber pressure according to the set value. The data acquisition and processing system includes a data acquisition unit 9 and a computer 10. The data acquisition unit 9 collects four measurement parameters: upstream stagnation pressure, upstream stagnation temperature, outlet back pressure, and gas mass flow rate, and transmits them to the computer 10. The computer 10 then performs subsequent theoretical calculations based on the collected boundary parameters.

[0014] Step 3: Using geometric parameters, thermophysical property parameters, total pressure at the pipe inlet, total temperature at the pipe inlet, and gas mass flow rate as inputs, simultaneously establish the control equations related to the compressible flow mass flow rate. Solve the control equations using a numerical iteration method to obtain the Mach number at the pipe inlet. Then, calculate the static pressure and static temperature at the pipe inlet based on the isentropic relationship.

[0015] The control equations related to the mass flow rate of compressible flow are established based on the flow characteristics of one-dimensional compressible adiabatic constant-section frictional pipe flow. These equations include isentropic relations, the equation of state for a complete gas, the definition of Mach number, the definition of sound velocity, and the definition of inlet mass flow rate. By simultaneously solving these equations, a closed nonlinear equation system can be formed. Substituting geometric parameters, thermophysical properties, and experimentally measured inlet total pressure, inlet total temperature, and gas mass flow rate into the equation system, the roots of this single-variable nonlinear equation can be obtained by solving the numerical iteration method, thus yielding the pipe inlet Mach number. The isentropic relations are fundamental formulas in compressible gas dynamics, establishing the correspondence between total parameters, static parameters, and Mach number. After obtaining the inlet Mach number, substituting it into the isentropic relations allows direct calculation of the pipe inlet static pressure and inlet static temperature. Simultaneously, substituting the Sutherland formula for calculating aerodynamic viscosity into the Reynolds number calculation formula yields the inlet Reynolds number under the given inlet total temperature, inlet total pressure, and mass flow rate, providing a basis for subsequent friction coefficient calculations. The specific process is as follows: (1) Ientropic relationship between inlet and outlet (1.1) (2) Equation of state for a complete gas (1.2) (3) Definition of Mach number (1.3) (4) Definition of speed of sound (1.4) (5) Definition of mass flow rate at the inlet Starting from the gas law, the definition of velocity, and the definition of the speed of sound, we have: (1.5) Combining equations 1.1-1.5, we obtain a closed system of equations: (1.6) Rearranging this system of nonlinear equations, we obtain its explicit form: (1.7) Input the total pressure at the pipe inlet measured by the experimental measurement system. Pipe inlet total temperature mass flow rate Solving for the terms on the right side of the equation yields the terms on the right side.

[0016] The roots of the above single-variable nonlinear equation, i.e., the entrance Mach number, are obtained by iterative methods. Then, the inlet static temperature can be obtained through the inlet isentropic relationship 1.1. Inlet static pressure .

[0017] Sutherland's formula for calculating aerodynamic viscosity Substitute into the Reynolds number calculation formula The Reynolds number at the inlet total temperature, inlet total pressure, and mass flow rate can be calculated.

[0018] Step 4: Calculate the critical outlet pressure when the outlet Mach number is 1 based on the inlet Mach number and the inlet static pressure. Compare the critical outlet pressure with the pipeline outlet back pressure to determine the congestion state of the flow in the pipeline and determine the corresponding outlet boundary conditions.

[0019] The critical outlet pressure is derived from the differential and integral relationships between static pressure and the square of Mach number in one-dimensional adiabatic frictional pipe flow theory. It is the outlet static pressure value corresponding to the Mach number reaching the limiting Mach number 1 (speed of sound) at the pipe outlet, and is a key threshold for determining whether the flow inside the pipe is congested. By combining the derived critical outlet pressure calculation formula with the inlet Mach number and inlet static pressure obtained in step 3, the specific value of the critical outlet pressure can be obtained. Comparing this critical outlet pressure with the outlet back pressure measured by the non-invasive experimental measurement system in step 2, the congestion state of the flow inside the pipe can be determined, and a unique outlet boundary condition can be determined based on different congestion states, providing clear constraints for the subsequent solution of outlet flow parameters, such as... Figure 3 As shown, this step is the core part of determining the flow state in the entire solution process.

[0020] Based on the differential and integral relationships (Equations 1.8-1.9) between static pressure and the square of Mach number in one-dimensional adiabatic frictional pipe flow theory, the ratio of static pressure at the pipe inlet / outlet is obtained as shown in Equation 1.10: (1.8) (1.9) (1.10) Setting the outlet Mach number to the limiting Mach number 1 (corresponding to the choke state), we obtain the formula for calculating the critical outlet pressure at the limiting Mach number: (1.11) Substituting the inlet Mach number and inlet static pressure, we obtain the critical outlet pressure at the limiting Mach number. This is compared with the outlet back pressure measured by the experimental measurement system. Comparison. If If this is the case, then the internal flow channel is considered to be blocked, and at this time, let ;like If so, the internal flow channel is considered to be in a non-blocked state, and at this time, let .

[0021] Step 5: Based on the outlet boundary conditions, the ratio of inlet / outlet static pressure and the isentropic relationship, the flow parameters at the outlet of the pipeline are obtained. The flow parameters at the outlet of the pipeline include the outlet Mach number, the outlet total pressure and the outlet static temperature.

[0022] The formula for the ratio of inlet / outlet static pressure is derived from the one-dimensional adiabatic frictional pipe flow theory, establishing a quantitative relationship between inlet / outlet static pressure, inlet / outlet Mach number, and specific heat ratio. This is used in step 4. and Substituting into Equation 1.10 yields the exit Mach number. Then, the outlet static temperature can be obtained using the isentropic relationship 1.1. Total export pressure Simultaneously, the outlet Reynolds number can be calculated using the same method, completing the solution for all flow parameters at the pipeline outlet. This provides the basic data at the outlet end for subsequent solutions of the average friction coefficient and friction parameters, such as... Figure 3 As shown, this step must be performed strictly in accordance with the congestion state boundary conditions determined in the previous steps.

[0023] Step 6: Based on the limiting pipe length theorem, the overall average friction coefficient is obtained by solving the inlet Mach number, outlet Mach number, pipe length, and hydraulic diameter.

[0024] The limiting pipe length theorem is a core theorem for one-dimensional adiabatic frictional pipe flow. Its core content is that for every subsonic inlet Mach number, there exists a maximum pipe length for which the outlet Mach number is exactly 1, i.e., the limiting pipe length. Based on integrating the friction coefficient and Mach number under the limiting state using the differential formula of the Mach number along the pipe length in one-dimensional adiabatic frictional pipe flow, the formula for calculating the limiting pipe length can be obtained. For flow with a fixed pipe length, the limiting pipe lengths for acceleration to the limiting Mach number 1 corresponding to the inlet and outlet Mach numbers respectively satisfy the relationship that the actual pipe length is the difference between the two limiting pipe lengths. Figure 4 The schematic diagram of the intermediate limit pipe length is shown. Substituting the inlet Mach number obtained in step 3, the outlet Mach number obtained in step 5, and the pipe length and hydraulic diameter obtained in step 1 into the above correlation formula, the overall average friction coefficient of the pipe can be obtained by solving the simultaneous equations. This average friction coefficient is a key input parameter for the subsequent fourth-order Runge-Kutta numerical integration to solve for the friction parameters. Specifically: According to the limiting tube length theorem (for every subsonic inlet Mach number, there exists a maximum tube length for which the outlet Mach number is exactly 1),... Based on the differential equation (Equation 1.12) of the Mach number along the pipe length in a one-dimensional adiabatic frictional pipe flow, integrating the friction coefficient and Mach number under the limiting state respectively, we obtain (Equation 1.13). (Fixed pipe length) The flow will , Each is considered as an inlet, and the corresponding limit pipe length satisfies Equation 1.14.

[0025] (1.12) (1.13) (1.14) in, , These represent the limiting tube lengths for acceleration to the ultimate Mach number, corresponding to the inlet and outlet Mach numbers, respectively. Combining equations 1.13 and 1.14, and substituting them into steps 3 and 5, we obtain... , The overall average friction coefficient is calculated based on the Mach numbers at the inlet and outlet (formula 1.15). (1.15) Step 7: Set the calculation step size, take the inlet Mach number as the initial condition for numerical integration, and combine the outlet Mach number and the average friction coefficient. Use the fourth-order Runge-Kutta method to iteratively solve the Reynolds number corresponding to each calculation step size, and perform correction calculations based on the friction coefficient calculation formula until the iteration conditions are met, and obtain the Mach number distribution along the process.

[0026] The calculation step size is a small length increment set along the pipe length direction for numerical integration. A smaller step size results in higher solution accuracy. The iteration condition is that the calculation position reaches the actual total length of the pipe. The inlet Mach number, as the initial condition for the fourth-order Runge-Kutta numerical integration, is one of the core technical features of this method. Starting from the pipe inlet (x=0), the Mach number is considered as a function of the pipe length to construct the flow differential equation. The fourth-order Runge-Kutta method is used to numerically integrate this flow differential equation. This method calculates the function values ​​at different intermediate points within each integration step and then performs a weighted average to obtain the Mach number at the next node, exhibiting high-precision solution characteristics. After completing the iterative solution of each calculation step, the Reynolds number at that position is calculated first. Then, the friction coefficient at that position is corrected based on the friction coefficient calculation formula. The corrected friction coefficient is fed back into the numerical integration of the next step, forming a coupled solution. This process is repeated until the calculation position reaches the total pipe length, ultimately obtaining a continuous distribution curve of the Mach number along the pipe length, i.e., the Mach number distribution along the pipe length, such as... Figure 3 As shown, this step is the core numerical calculation step for solving the friction parameters.

[0027] Based on the fourth-order Runge-Kutta method, the initialization, numerical integration, parameter derivation, result verification, and output stages are performed. The specific process is as follows: 1. Initialization: Starting from the pipe inlet ( ), set initial conditions: (1.16) Treat the solution quantity as The function, let The equation is obtained. .

[0028] 2. Numerical Integration: The fourth-order Runge-Kutta method (RK4) is used to numerically integrate the flow differential equation. The RK4 method calculates the function values ​​at different intermediate points within each integration step and then calculates a weighted average to obtain the function value for the next node. The solution formula is as follows: (1.17) After one calculation step, the Reynolds number at that location is calculated and corrected based on the friction coefficient calculation formula (Equation 1.18). This process is repeated until... (Pipe length), obtain the Mach number distribution along the path. .

[0029] (1.18) Step 8: Based on the gas dynamics control volume method, iteratively solve for the distribution curves of static pressure, velocity, static temperature and density along the path based on the known Mach number distribution.

[0030] The gas dynamics control volume method is derived based on the three conservation laws of mass, energy, and momentum. Specifically, it uses the logarithmic differential form of the mass conservation equation, the differential form of the energy equation for steady gas flow, and the differential form of the momentum equation, treating Mach number as an independent variable. This results in equations relating Mach number to differential components of static pressure, velocity, static temperature, and density. Since the pipe is a straight pipe with a uniform cross-section, the differential component of the cross-sectional area is zero, allowing for simplification. Based on the Mach number distribution along the pipe length obtained in step 7, the Mach number at each calculation step is substituted into the simplified differential component equations. The same numerical iteration method as in step 7 is used to solve the problem, sequentially obtaining the values ​​of static pressure, velocity, static temperature, and density for each calculation step. This generates continuous distribution curves of these parameters along the pipe length, completing the solution for the pipe length distribution of all core flow parameters in a one-dimensional uniform cross-section adiabatic friction pipe flow. Figure 3 As shown, this step achieves the simultaneous solution of multiple parameters based on the Mach number distribution along the path, specifically as follows: Numerical integration: Referring to the control volume method in gas dynamics, based on the logarithmic differential form of the mass conservation equation, and the differential forms of the energy equation and momentum equation for steady flow of a perfectly stable gas, the integration is performed... As independent variables, they are arranged into equations relating to the other five differential components: (1.19) Given the Mach number, the static pressure is calculated using the same iterative method. ,speed , static temperature With density Complete distribution curves of parameters along the path.

[0031] Step 9: Compare and verify the predicted static pressure at the pipeline outlet with the back pressure at the outlet measured in Step 2, and output the inlet section parameters, friction distribution curve, average friction coefficient, and verification error.

[0032] The comparison and verification involves comparing the predicted static pressure at the pipe outlet obtained in step 8 with the measured back pressure at the outlet obtained in step 2 using a non-invasive experimental measurement system. The relative error between the two values ​​is calculated. If the error is within a preset range, the verification is complete, validating the solution accuracy of this method. The output includes inlet section parameters, friction distribution curves for each flow parameter, the overall average friction coefficient of the pipe, and the verification error. It also includes all flow parameters at the pipe outlet section. The output friction distribution curves are continuous curves, fully reflecting the evolution of each flow parameter along the pipe length. Figure 3 As shown, this step is the final stage of the entire solution process, realizing the verification and output of the solution results. Specifically: The predicted outlet static pressure value calculated in the previous step The outlet back pressure obtained from step 2 The accuracy of the method is verified through comparison. The output results include inlet and outlet cross-sectional parameters, friction distribution curves, average friction coefficient, and verification error.

[0033] The aforementioned method for solving the flow parameters along a one-dimensional adiabatic friction pipe with uniform cross-section can obtain the flow parameter distribution and average friction coefficient of a one-dimensional adiabatic friction pipe flow within a closed, uniform cross-section channel with high precision, based on the measurement of the physical parameters at the pipe inlet / outlet. This method overcomes the bottlenecks of traditional measurement and simulation caused by the closed structure and small scale, thus providing direct and reliable data for evaluating the flow parameter distribution along the channel without damaging the channel structure or interfering with the actual flow. This significantly improves the efficiency and engineering practicality of the design, analysis, and testing of cooling medium delivery pipelines for high-speed aircraft.

[0034] In one embodiment, the compressible flow mass flow rate related control equations include isentropic relations, the equation of state for a complete gas, the definition of Mach number, the definition of sound speed, and the definition of inlet mass flow rate.

[0035] Specifically, the isentropic relationship is the core fundamental formula for isentropic flow of compressible gases. It establishes quantitative relationships between total pressure and static pressure and Mach number, as well as between total temperature and static temperature and Mach number. It is applicable to the mutual conversion between total parameters and static parameters at the inlet and outlet of the pipeline in this method. It is the key formula for solving the static pressure, static temperature, and Mach number at the inlet and outlet. Among them, total pressure and total temperature are parameters directly measured in experiments, static pressure and static temperature are flow state parameters that need to be solved, and Mach number is the core dimensionless flow parameter.

[0036] In one embodiment, geometric parameters, thermophysical property parameters, and inlet total pressure, inlet total temperature, and gas mass flow rate are used as inputs to simultaneously establish control equations related to the compressible flow mass flow rate, including: Using geometric parameters, thermophysical properties, inlet total pressure, inlet total temperature, and gas mass flow rate as inputs, the simultaneous control equations related to the compressible flow mass flow rate are: ; in, Indicates the inlet volumetric flow rate. Indicates the inlet fluid density. Indicates the inlet fluid velocity. Represents the cross-sectional area of ​​the pipe. Indicates the total inlet pressure. Indicates the inlet static pressure. Indicates the Mach number at the entrance. Indicates the total inlet temperature. Indicates the inlet static temperature. This represents the gas constant.

[0037] After rearranging the control equations related to the mass flow rate of compressible flow, the explicit form of the control equations related to the mass flow rate of compressible flow is obtained as follows: ; in, This indicates the gas mass flow rate.

[0038] Specifically, starting from the complete gas equation of state, the definition of velocity, and the definition of sound velocity, the five governing equations mentioned above are simultaneously derived and rearranged. This eliminates intermediate variables such as inlet fluid density, inlet fluid velocity, inlet static pressure, and inlet static temperature, ultimately yielding an explicit nonlinear equation with only the inlet Mach number as the unknown. Substituting the experimentally measured inlet total pressure, inlet total temperature, gas mass flow rate, and known pipe cross-sectional area, gas constant, and specific heat ratio into this explicit equation, and solving the roots of the equation using a numerical iteration method, the inlet Mach number can be obtained. This simultaneous solution process provides a complete theoretical basis and calculation method for solving the core parameter of the inlet Mach number, and is the foundation for solving all subsequent parameters.

[0039] In one embodiment, the critical outlet pressure at which the outlet Mach number is 1 is determined based on the inlet Mach number and the inlet static pressure, including: The critical outlet pressure when the outlet Mach number is 1 is calculated based on the inlet Mach number and the inlet static pressure: ; in, Indicates the inlet static pressure. Indicates the Mach number at the entrance. It indicates the specific heat ratio.

[0040] In one embodiment, the critical outlet pressure is compared with the outlet back pressure to determine the congestion state of the flow in the pipeline and to determine the corresponding outlet boundary conditions, including: The critical outlet pressure is compared with the outlet back pressure measured by a non-invasive experimental measurement system. In comparison, if If this is the case, then the internal flow channel is considered to be blocked, and at this time, let ;like If so, the internal flow channel is considered to be in a non-blocked state, and at this time, let ,in, Indicates the outlet static pressure. Indicates the Mach number for exports.

[0041] Specifically, the choked state is a special flow state of one-dimensional adiabatic frictional pipe flow. When the outlet back pressure is lower than the critical outlet pressure, the outlet Mach number reaches the speed of sound, and the flow rate no longer increases with decreasing outlet back pressure. At this point, the outlet Mach number is fixed at 1, which is the only boundary condition. When the outlet back pressure is higher than or equal to the critical outlet pressure, the flow inside the pipe is non-choked and subsonic. At this point, the outlet static pressure equals the measured outlet back pressure, which is the only boundary condition. This judgment method is based on quantitative numerical comparison, and the result is unique and clear. It provides definite constraints for the subsequent solution of outlet flow parameters, avoids the problem of multiple solutions in the solution process, and is a key link to ensure the uniqueness of the solution result.

[0042] In one embodiment, the pipe outlet flow parameters are obtained by solving based on the outlet boundary conditions, the ratio of pipe inlet / outlet static pressure, and the isentropic relationship, including: Substitute the outlet static pressure and the outlet Mach number to be solved into the formula for the ratio of pipe inlet / outlet static pressure to obtain the outlet Mach number, and then calculate the outlet static temperature and outlet total pressure using the isentropic relationship. The formula for the ratio of the static pressure at the inlet / outlet of the pipeline is: ; in, Indicates the outlet static pressure. Indicates the inlet static pressure. Indicates the Mach number at the entrance. Indicates the export Mach number. It indicates the specific heat ratio.

[0043] In one embodiment, the overall average friction coefficient is obtained by solving for the inlet Mach number, outlet Mach number, pipe length, and hydraulic diameter based on the limiting pipe length theorem, including: The process of obtaining the overall average friction coefficient based on the limiting pipe length theorem, using the inlet Mach number, outlet Mach number, pipe length, and hydraulic diameter, is as follows: ; in, , These represent the limiting tube lengths for acceleration to the ultimate Mach number, corresponding to the inlet and outlet Mach numbers, respectively. The overall average coefficient of friction, The hydraulic diameter, This is the actual length of the pipe. It is a universal function of Mach number. Indicates the Mach number at the entrance. Indicates the Mach number for exports.

[0044] Specifically, the solution process is based on the limiting pipe length theorem, which fully considers the intrinsic relationship between Mach number, pipe length, and friction coefficient in one-dimensional adiabatic friction pipe flow. The average friction coefficient obtained reflects the frictional resistance characteristics of the entire pipe and is a key input parameter for subsequent fourth-order Runge-Kutta numerical integration to solve the friction parameters along the pipe, providing a basis for the correction of the friction coefficient along the pipe.

[0045] In one embodiment, the fourth-order Runge-Kutta method is used to iteratively solve for the Reynolds number corresponding to each computational step, including: When using the fourth-order Runge-Kutta method, the Mach number is considered as the pipe length. The function, let Construct flow differential equations The solution formula for the fourth-order Runge-Kutta method is: ; in, , , , Both represent the intermediate slope term. Indicates the step size for numerical integration. This represents the result after one iteration.

[0046] In one embodiment, the process of correcting the calculation based on the friction coefficient calculation formula is as follows: ; in, Represents the Reynolds number.

[0047] Within each calculation step, the corresponding flow regime is determined based on the obtained Reynolds number, and then the friction coefficient at that location is calculated by substituting it into the corresponding formula, thereby realizing the friction coefficient correction along the flow path and making the solution result more consistent with the actual flow characteristics.

[0048] In one embodiment, based on the gas dynamics control volume method, the friction curves of static pressure, velocity, static temperature, and density are iteratively solved based on the known Mach number friction curves, including: The gas dynamics control volume method is based on the logarithmic differential form of the mass conservation equation, the differential form of the energy equation for steady gas flow, and the differential form of the momentum equation. Using the Mach number as an independent variable, it constructs the following differential equation relationships: ; in, Represents the Mach number. Indicates the coefficient of friction. Indicates specific heat ratio. Indicates static pressure. Indicates the hydraulic diameter. Indicates flow rate, Indicates density, Indicates static temperature.

[0049] Specifically, based on the Mach number distribution along the pipe obtained in step 7, the Mach number of each calculation step, the corrected friction coefficient, and the known specific heat ratio and hydraulic diameter are substituted into the above equation to solve for the relative differential components of each parameter. Then, combined with the initial values ​​of static pressure, velocity, static temperature, and density at the pipe inlet, the actual values ​​of the parameters corresponding to each calculation step are obtained through integral calculation, thereby generating continuous distribution curves of static pressure, velocity, static temperature, and density along the pipe length. This realizes the joint solution of multiple flow parameters along the pipe and completely reconstructs the parameter distribution along the pipe of one-dimensional uniform cross-section adiabatic friction pipe flow.

[0050] It should be understood that, although Figure 1 The steps in the flowchart are shown sequentially as indicated by the arrows, but these steps are not necessarily executed in the order indicated by the arrows. Unless otherwise specified herein, there is no strict order in which these steps are executed, and they can be performed in other orders. Figure 1 At least some of the steps in the process may include multiple sub-steps or multiple stages. These sub-steps or stages are not necessarily completed at the same time, but can be executed at different times. The execution order of these sub-steps or stages is not necessarily sequential, but can be executed in turn or alternately with other steps or at least some of the sub-steps or stages of other steps.

[0051] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0052] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these modifications and improvements all fall within the protection scope of this application. Therefore, the protection scope of this application should be determined by the appended claims.

Claims

1. A method for solving friction parameters of one-dimensional adiabatic friction pipe flow with uniform cross-section, characterized in that, The method includes: Step 1: Obtain the geometric parameters of the pipe to be tested and the thermophysical properties of the working fluid inside the pipe. The geometric parameters include cross-sectional area, length, and hydraulic diameter. The thermophysical properties include gas constant and specific heat ratio. Step 2: Collect the total pressure at the pipe inlet, the total temperature at the pipe inlet, the back pressure at the pipe outlet, and the mass flow rate of the gas flowing through the pipe using a non-invasive experimental measurement system; Step 3: Using geometric parameters, thermophysical property parameters, total pressure at pipe inlet, total temperature at pipe inlet, and gas mass flow rate as inputs, simultaneously establish the control equations related to the compressible flow mass flow rate. Solve the control equations using numerical iteration to obtain the Mach number at pipe inlet. Then, calculate the static pressure and static temperature at pipe inlet based on the isentropic relationship. Step 4: Calculate the critical outlet pressure when the outlet Mach number is 1 based on the inlet Mach number and the inlet static pressure. Compare the critical outlet pressure with the pipeline outlet back pressure to determine the congestion state of the flow in the pipeline and determine the corresponding outlet boundary conditions. Step 5: Based on the outlet boundary conditions, the ratio of inlet / outlet static pressure and the isentropic relationship, the flow parameters at the outlet of the pipeline are obtained. The flow parameters at the outlet of the pipeline include the outlet Mach number, the outlet total pressure and the outlet static temperature. Step 6: Based on the limiting pipe length theorem, the overall average friction coefficient is obtained by solving for the inlet Mach number, outlet Mach number, pipe length, and hydraulic diameter. Step 7: Set the calculation step size, take the inlet Mach number as the initial condition for numerical integration, combine the outlet Mach number and the average friction coefficient, and use the fourth-order Runge-Kutta method to iteratively solve the Reynolds number corresponding to each calculation step size. Then, make correction calculations based on the friction coefficient calculation formula until the iteration conditions are met and the Mach number distribution along the process is obtained. Step 8: Based on the gas dynamics control volume method, iteratively solve the flow distribution curves of static pressure, velocity, static temperature and density based on the known Mach number flow distribution; Step 9: Compare and verify the predicted static pressure at the pipeline outlet with the back pressure at the outlet measured in Step 2, and output the inlet section parameters, friction distribution curve, average friction coefficient, and verification error.

2. The method according to claim 1, characterized in that, The compressible flow mass flow rate related control equations include isentropic relations, the equation of state for a complete gas, the definition of Mach number, the definition of sound speed, and the definition of inlet mass flow rate.

3. The method according to claim 2, characterized in that, Using geometric parameters, thermophysical properties, and the total pressure, total temperature, and mass flow rate at the pipe inlet as inputs, simultaneously establish the control equations related to the compressible flow mass flow rate, including: Using geometric parameters, thermophysical properties, total inlet pressure, total inlet temperature, and gas mass flow rate as inputs, the following simultaneous control equations related to compressible flow mass flow rate are established: in, Indicates the inlet volumetric flow rate. Indicates the inlet fluid density. Indicates the inlet fluid velocity. Represents the cross-sectional area of ​​the pipe. This indicates the total pressure at the pipe inlet. Indicates the inlet static pressure. Indicates the Mach number at the entrance. Indicates the total temperature at the pipe inlet. Indicates inlet static temperature. Represents the gas constant; After rearranging the control equations related to the mass flow rate of compressible flow, the explicit form of the control equations related to the mass flow rate of compressible flow is obtained as follows: in, This indicates the gas mass flow rate.

4. The method according to claim 1, characterized in that, The critical outlet pressure at which the outlet Mach number is 1 is calculated based on the inlet Mach number and the inlet static pressure, including: The critical outlet pressure when the outlet Mach number is 1 is calculated based on the inlet Mach number and the inlet static pressure: in, Indicates the inlet static pressure. Indicates the Mach number at the entrance. It indicates the specific heat ratio.

5. The method according to claim 4, characterized in that, By comparing the critical outlet pressure with the pipeline outlet back pressure, the congestion state of the flow within the pipeline is determined, and the corresponding outlet boundary conditions are identified, including: The critical outlet pressure is compared with the outlet back pressure measured by a non-invasive experimental measurement system. In comparison, if If this is the case, then the internal flow channel is considered to be blocked, and at this time, let ;like If so, the internal flow channel is considered to be in a non-blocked state, and at this time, let ,in, Indicates the outlet static pressure. Indicates the Mach number for exports.

6. The method according to claim 5, characterized in that, Based on the aforementioned outlet boundary conditions, the ratio of inlet / outlet static pressure, and the isentropic relationship, the flow parameters at the pipe outlet are obtained, including: Substitute the outlet static pressure and the outlet Mach number to be solved into the formula for the ratio of pipe inlet / outlet static pressure to obtain the outlet Mach number, and then calculate the outlet static temperature and outlet total pressure using the isentropic relationship. The formula for the ratio of the static pressure at the inlet / outlet of the pipeline is: in, Indicates the outlet static pressure. Indicates the inlet static pressure. Indicates the Mach number at the entrance. Indicates the export Mach number. It indicates the specific heat ratio.

7. The method according to claim 1, characterized in that, The overall average friction coefficient is obtained by solving the limit pipe length theorem based on the inlet Mach number, outlet Mach number, pipe length, and hydraulic diameter, including: The process of obtaining the overall average friction coefficient based on the limiting pipe length theorem, using the inlet Mach number, outlet Mach number, pipe length, and hydraulic diameter, is as follows: in, , These represent the limiting tube lengths for acceleration to the ultimate Mach number, corresponding to the inlet and outlet Mach numbers, respectively. The overall average coefficient of friction, The hydraulic diameter, This is the actual length of the pipe. It is a universal function of Mach number. Indicates the Mach number at the entrance. Indicates the Mach number for exports.

8. The method according to claim 1, characterized in that, The Reynolds number for each computational step is solved iteratively using the fourth-order Runge-Kutta method, including: When using the fourth-order Runge-Kutta method, the Mach number is considered as the pipe length. The function, let Construct flow differential equations The solution formula for the fourth-order Runge-Kutta method is: in, , , , Both represent the intermediate slope term. Indicates the step size for numerical integration. This represents the result after one iteration.

9. The method according to claim 1, characterized in that, The process of correcting the calculation based on the friction coefficient formula is as follows: in, Represents the Reynolds number.

10. The method according to claim 1, characterized in that, Based on the gas dynamics control volume method, the friction curves of static pressure, velocity, static temperature, and density are obtained iteratively from the known Mach number friction curves, including: The gas dynamics control volume method is based on the logarithmic differential form of the mass conservation equation, the differential form of the energy equation for steady gas flow, and the differential form of the momentum equation. Using the Mach number as an independent variable, it constructs the following differential equation relationships: in, Represents the Mach number. Indicates the coefficient of friction. Indicates specific heat ratio. Indicates static pressure. Indicates the hydraulic diameter. Indicates flow rate, Indicates density, Indicates static temperature.

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