Method for predicting pneumatic hammer of pipeline

[0063] The technical problem solved by the present invention is to provide a Lagrangian particle method for solving the problem of natural gas pipeline transmission. The method adopts the smooth particle hydrodynamic method to solve the shock tube physics equation under the moving coordinate system, and fully considers the strength of the gas. The impact of compressibility can more easily simulate the impact of gas on the pipeline under the premise of satisfying numerical accuracy.

[0064] Step 1: Initialization: Initialize the relevant variables and particle information of the system;

[0065] Step two, generate particle information;

[0066] Step three, list the equations to be solved and iteratively calculate:

[0067] According to the principle of the natural gas pipeline problem, the mathematical model of the physical model can be derived, that is, the control equation in Lagrangian form is:

[0068]

[0069]

[0070]

[0071] Where P is the gas pressure, ν is the gas velocity, ρ is the gas density, and e is the gas energy. The Lagrangian governing equations in (1)~(3) are not closed, and state equations need to be added to ensure that the system is complete.

[0072] Holistic. For an ideal gas, the equation of state is:

[0073] p=(γ-1)ρe (4)

[0074] For the air hammer problem in the actual pipeline, if the temperature changes little, the problem can be simplified, and the corresponding state equation is:

[0075]

[0076] At this time, the energy term will be decoupled from the governing equation.

[0077] In order to solve the above equations and predict the air hammer problem in natural gas pipelines, the present invention proposes a Lagrangian particle method. The details are as follows:

[0078] In the smooth particle hydrodynamics method, the integral expression of the function f(x) is:

[0079] f(x)=∫ Ω f(x′)δ(x-x′)dx′ (6)

[0080] Where δ(x-x') is the Dirac function, and Ω is the integral volume containing x. If the Dirac function is replaced by the smooth function W(x-x′,h), the integral expression of f(x) is:

[0081] f(x)≈∫ Ω f(x′)W(x-x′,h)dx′ (7)

[0082] The function derivative integral is expressed as:

[0083]

[0084] also because

[0085]

[0086] and so

[0087]

[0088] Then the particle approximation gets:

[0089]

[0090] because

[0091]

[0092] and so

[0093]

[0094] Again

[0095]

[0096] Therefore, the particle approximation of the function at particle i can be written as:

[0097]

[0098] Using smooth particle hydrodynamics method to discretize equations (1)~(3), we can get equations (16)~(18)

[0099]

[0100]

[0101]

[0102] p=(γ-1)ρe (19)

[0103]

[0104] Where Π ij Monaghon type artificial viscosity, H ij It is an artificial heat term, and γ and ξ are constant coefficients.

[0105] According to equations (16), (17) and (18), the density, velocity and energy information of each particle at different times are calculated, and then the pressure of the particles can be calculated according to equation (19). The specific calculation process is as follows:

[0106] The specific calculation process is:

[0107] 6) Loop each time step;

[0108] 7) After initializing the attribute information of all particles in the calculation domain, search for the neighboring particles of the target particle to obtain the initial value of velocity and mass, and then calculate the derivative of the particle's density through equation (16), and finally update the particle density through Euler time integration information;

[0109] 8) The velocity derivative is calculated through the updated density information obtained in the previous step, the initial pressure value and the Monaghon type artificial viscosity equation, and then the velocity information of the particles is updated through Euler time integration, and finally the position of the particles is updated through the velocity information;

[0110] 9) According to the density and velocity particle information obtained in the above two steps and the Monaghon type (a method proposed by Monaghon to solve numerical oscillations in 1992), the energy derivative is calculated by the artificial viscosity equation and the artificial heat term equation, and then the particles are updated through Euler time integration Energy information;

[0111] 10) Combining equation (19) with the density and energy information obtained above, update the particle pressure;

[0112] 6) Then update the smooth length information of the particle search by formula (20).

[0113] Step four, output the result:

[0114] 1) At the end of each time step, save the intermediate results, and output the intermediate results;

[0115] 2) End the time loop and output the final result.

[0116] Further, in the above solution, the initialization variable information and operating parameters are specifically set as follows:

[0117] CASE 1: The left side of the natural gas pipeline impacts the right side expansion problem

[0118] See the physical model of the simulation problem in this experiment figure 2 , The calculation domain is a one-dimensional space with a length of 30m, both sides of the pipeline are open, the left end is set as the inlet, the right end is set as the outlet, and a partition is set in the middle of the pipeline to simulate the valve in the natural gas pipeline. At the initial moment, there are two different states of gas on both sides of the partition, and the experimental problem is simulated by opening the middle partition. In this experiment, we added virtual particles at the entrance and exit to make up for the lack of boundary particles.

[0119] In the CASE1 problem, our initial conditions are set as: P L = 7, V L = 0,ρ L = 1, P R =10,ρ R = 1,

[0120] V R =0. In this experiment, we arrange 1000 particles of the same mass on the left end of the pipe, and 1000 particles of the same mass on the right end of the pipe. The experimental simulation time is 1s, and the time integration step is 10e-5s. The parameters of Monaghon type artificial viscosity are α=1, β=1; the coefficients of the artificial heat term are g1=0.2, g2=0.4.

[0121] Based on the SPH discrete scheme (15)~(17) of the pipeline equation, the Lagrangian particle method for simulating one-dimensional natural gas pipeline is:

[0122] Step one, initialization. Initialize the relevant variables and operating parameters of the system, including:

[0123] Such as figure 1 As shown, the natural gas pipeline length L is 30m, and the pressure, velocity, and density at the left end of the pipeline at 0~15m are: 7Pa, 0m/s, 1kg/m 3 , 15~30m on the right are respectively: 10Pa, 0m/s, 1kg/m 3. Particle spacing Δx 0 =0.01m, the calculation time step is 10e-5s, and the calculation time is 1s. In this experiment, the variable smooth length is used for calculation, the initial smooth length h 0 = 1.5Δx. Take the cubic spline function as the kernel function.

[0124] Step two, generate particle information, including:

[0125] In the step of initializing particles, a total of 2000 gas particles are generated (not including virtual particles). The virtual particles are arranged on the initial boundaries of the left and right sides of the calculation domain to ensure that the gas particles within 2h from the left and right measurement initial positions can perform normal operations. The density, velocity, pressure and energy of the particles are the same as those of gas particles.

[0126] Step three, list the equations to be solved and iteratively calculate:

[0127]

[0128]

[0129]

[0130] Where P is the gas pressure, ν is the gas velocity, ρ is the gas density, and e is the gas energy. The Lagrangian governing equations in (1)~(3) are not closed, and state equations need to be added to ensure the integrity of the system. For an ideal gas, the equation of state is:

[0131] p=(γ-1)ρe (4)

[0132] For the air hammer problem in the actual pipeline, if the temperature changes little, the problem can be simplified, and the corresponding state equation is:

[0133]

[0134] At this time, the energy term will be decoupled from the governing equation.

[0135] In the smooth particle hydrodynamics method, the integral expression of the function f(x) is:

[0136] f(x)=∫ Ω f(x′)δ(x-x′)dx′ (6)

[0137] Where δ(x-x') is the Dirac function, and Ω is the integral volume containing x. If the Dirac function is replaced by the smooth function W(x-x′,h), the integral expression of f(x) is:

[0138] f(x)≈∫ Ω f(x′)W(x-x′,h)dx′ (7)

[0139] The function derivative integral is expressed as:

[0140]

[0141] also because

[0142] and so

[0143]

[0144]

[0145] Then the particle approximation gets:

[0146]

[0147] because

[0148]

[0149] and so

[0150]

[0151] Again

[0152]

[0153] Therefore, the particle approximation of the function at particle i can be written as:

[0154]

[0155] Using smooth particle hydrodynamics method to discretize equations (1)~(3), we can get equations (16)~(18)

[0156]

[0157]

[0158]

[0159] p=(γ-1)ρe (19)

[0160]

[0161] Where Π ij Monaghon type artificial viscosity, H ij It is an artificial heat term, and γ and ξ are constant coefficients.

[0162] According to equations (16), (17) and (18), the density, velocity and energy information of each particle at different times are calculated, and then the pressure of the particles can be calculated according to equation (19). The specific calculation process is as follows:

[0163] The specific calculation process is:

[0164] 1) Loop each time step;

[0165] 2) After initializing the attribute information of all particles in the calculation domain, search for the neighboring particles of the target particle to obtain the initial value of velocity and mass, and then calculate the derivative of the particle's density through equation (16), and finally update the particle density through Euler time integration information;

[0166] 3) The velocity derivative is calculated through the updated density information obtained in the previous step, the initial pressure value and the Monaghon-type artificial viscosity equation, and then the velocity information of the particles is updated through Euler time integration, and finally the position of the particles is updated through the velocity information;

[0167] 4) Calculate the energy derivative based on the density and velocity particle information obtained in the above two steps, the Monaghon type artificial viscosity equation and the artificial heat term equation, and then update the energy information of the particles through Euler time integration;

[0168] 5) Combining equation (19) with the density and energy information obtained above, update the particle pressure;

[0169] 6) Then update the smooth length information of the particle search by formula (20).

[0170] Step four, output the result:

[0171] 1) At the end of each time step, save the intermediate results, and output the intermediate results;

[0172] 2) End the time loop and output the final result.

[0173] CASE 2: Pipeline air hammer problem

[0174] See the physical model of the simulation problem in this experiment Figure 4 , The natural gas pipeline length L is 37.5m, the pressure inside the gas tank is P R 250Kpa, the initial velocity of the gas in the pipe is 10m/s, the pipe diameter D is 0.0221m, and the gas density ρ is 1.2kg/m 3 , The friction coefficient f is 0.02. The initial condition of natural gas transmission is V(x,0)=V 0 with 0

[0175] In an example, the specific steps are as follows:

[0176] Step 1. Initialization: Initialize related variables and particle (including added virtual particles) information. Specifically:

[0177] 1) Initialize the variable information related to the problem: initialize the variable information, the diameter of the pipeline D is 0.0221m, the length of the natural gas pipeline L is 37.5m, and the pressure P in the gas tank R Is 250Kpa, the density of gas is 1.2kg/m 3 , The initial velocity of the gas in the pipeline is 10m/s, the calculation time step is 0.0001s, and the total calculation time is 46.524s, etc.;

[0178] 2) Initialize the gas particle information, distribute the particles uniformly in the fluid domain, and add the initial information: initialize the fluid particles, the pipeline uniformly distributes a total of 372 gas particles, and the gas particle information is V(x,0)=V 0 with 0

[0179] 3) Initialize the virtual particle information, place two layers of virtual particles on the upstream and downstream boundaries of the fluid, and add initial information according to the boundary conditions: two virtual particles at the upstream and downstream boundaries, and the upstream virtual particle pressure is P R , The downstream virtual particle pressure is 0, and the initial velocity of the upstream and downstream virtual particles is V 0.

[0180] Step two, list the equations to be solved and iteratively calculate, the steps are the same as step three in CASE1, replace the equation of state with equation (5).

[0181] Step three is the same as step four in CASE1.

[0182] Although the present invention has been described above with reference to the drawings, the present invention is not limited to the above-mentioned specific embodiments. The above-mentioned specific embodiments are only illustrative and not restrictive. Those of ordinary skill in the art are Under the enlightenment, many modifications can be made without departing from the purpose of the present invention, and these all fall within the protection of the present invention.