A simulation method for large-scale water transmission systems incorporating gas dynamic characteristics

Abstract

The present invention discloses a simulation method for large-scale water delivery systems that incorporates the dynamic characteristics of gases. The method derives the continuity and momentum equations of the gas, constructing correlations between density, wave velocity, flow rate, and parameters representing the dynamic characteristics of the gas. The method uses the characteristic line method to transform the quasi-linear hyperbolic partial differential equation governing the gas, converting it into an ordinary differential equation and solving the eigenvalues. Finally, the water and gas total differential equations are derived simultaneously at the water-gas interface to solve the entire water-filled pipeline. The present invention develops a pressurized elastic water column filling model that incorporates the dynamic characteristics of gas and water for the rapid filling process. The simulation results of the proposed model that incorporates gas characteristics are compared with those of a model that considers the correlation between gas pressure and volume changes. The former model can reflect the differences in pressure changes at different nodes within the gas, while the latter model only assumes that the pressure is consistent across all regions within the gas. Comparison of the simulation results of the two models with experimental data shows that the gas-water transient flow model that incorporates gas characteristics is more accurate, especially in large-scale pipeline systems. Models that do not consider gas characteristics tend to underestimate the pressure peak during pressure transients.

Description

Technical Field

[0001] The present invention relates to numerical simulation calculation technology in the field of hydraulics of hydropower stations (pump stations), and specifically to a simulation method suitable for incorporating gas dynamic characteristics into large-scale water transmission systems, targeting the transient flow process of gas and water in long-distance water transmission pipelines. Background Art

[0002] Developing a robust water system is crucial to addressing water resource inequality and promoting national economic growth. The immense length and complex design of long-distance water transmission systems present unique challenges. Pipelines undergo changes in operating conditions, filling, and draining processes during maintenance and repair, which can result in complex air-water two-phase flows with large amounts of entrained air. The presence of air can either mitigate the effects of pressure waves or create extreme, destructive pressures, leading to catastrophic pipe bursts or damage to facilities. In addition, entrained air reduces the effective flow area of ​​the pipeline, thereby reducing engineering benefits. Therefore, it is essential to accurately simulate the transient pressures during the rapid filling of large, long-distance water transmission pipelines.

[0003] Currently, one-dimensional models for simulating fast-filling pipeline problems are generally divided into two categories:

[0004] The first type applies pressurized flow equations and open flow equations to different flow regions respectively, which are collectively called interface tracking mixing flow models;

[0005] The second is to use a set of equations to simulate the pressurization effect of the flow area.

[0006] These models often ignore gas properties or assume that air is homogeneous and variable. Due to a lack of consideration for the discretization of gas space, they cannot fully reflect the dynamic characteristics of gas compression and expansion, which is a knowledge gap in existing models. Summary of the Invention

[0007] To address the inaccurate simulation of gas-water transient flows in large-scale water conveyance systems due to neglect of gas dynamics, this paper provides a simulation method suitable for large-scale water conveyance systems that incorporates gas dynamics. The governing equations for air are introduced and combined with the governing equations for water during rapid filling and the interfacial continuity equation to derive the proposed model that incorporates air properties. This approach addresses the knowledge gap regarding appropriate methods for incorporating pressurized air properties into numerical model development. A pressurized elastic water column filling model that accounts for air effects is developed and compared with a uniform air model and experimental results to evaluate the accuracy of the proposed model.

[0008] The present invention provides a simulation method for incorporating gas dynamic characteristics into a large-scale water delivery system, comprising the following steps:

[0009] S1: Derive the continuity equation and momentum equation of the gas, and construct the correlation relationship between the density, wave velocity, and flow rate parameters that represent the dynamic characteristics of the gas;

[0010] S2: Use the characteristic line method to transform the gas control quasi-linear hyperbolic partial differential equation into an ordinary differential equation and solve the eigenvalue;

[0011] S3: After deriving the water and gas total differential equations simultaneously at the water-gas interface, the full domain solution of the water-filled pipe is obtained.

[0012] Furthermore, in the derivation of the gas continuity and momentum control equations in step S1, the continuity and momentum equations of the air in the derivation process can be written as:

[0013]

[0014] Where, subscript g is gas; ρ g is the air density; u g is the air speed; P g It is the air pressure.

[0015] The relationship between air wave speed and air density is as follows:

[0016] c g 2 =P g ρ g -1 (3)

[0017]

[0018] Where c g is the air wave speed; m is the constant multidirectional index; c g0 ,P g0 It is c g and P g The initial value of .

[0019] The following equation (5) is derived from the joint solution of equations (3) and (4).

[0020]

[0021] Where k is a constant,

[0022] Substituting Equation (5) into Equations (3) and (4), we can obtain the derived gas continuity and momentum control equations:

[0023]

[0024] Furthermore, in the transformation process of the gas control quasi-linear hyperbolic partial differential equation in step S2, for the water-filled pipe section, the positive and negative characteristic lines of the standard one-dimensional unsteady flow are integrated to obtain the following formula:

[0025]

[0026] Where i is the location of the current solution node, t+Δt is the next solution time, D is the pipe diameter, A is the pipe area, and subscript w is water; H w is the pressure measuring head; a w is the wave velocity; G is the acceleration due to gravity; Δx is the unit distance along the pipe.

[0027] The air momentum equation and the continuity equation (6) and equation (7) form a pair of quasi-linear hyperbolic partial differential equations, with the two dependent variables being the velocity u and g and air density c g , the two independent variables are the distance x on the pipeline and the time t. The equation is transformed into an ordinary differential equation using the characteristic line. The simplified air control motion equation and continuity equation can be identified as L1 and L2 respectively from Equations (8) and (9):

[0028]

[0029] Furthermore, in step S2, these equations are linearly combined using an unknown multiplier λ:

[0030]

[0031] Any two real numbers, with different input values, will again produce two dependent variables u g and c g The two equations are equivalent in all respects. Equations (10) and (11) can be simplified to equation (12) by properly selecting two specific values ​​of λ. In general, the variable u g and c g is a function of x and t, then, from calculus:

[0032]

[0033] Combining equations (11) and (13), we can see that:

[0034]

[0035] Equation (12) becomes an ordinary differential equation:

[0036]

[0037] The solution of Equation (14) yields two special values ​​of λ:

[0038]

[0039] Furthermore, in step S2, the wave propagation speed c g and speed u g It shows the relationship between the position change of the wave and the time change. When the positive value is taken in equation (16), the positive value of λ must be taken in equation (15). Similar parallelism exists for negative λ. Substituting these values ​​of λ into equation (15) yields two pairs of equations. This pressure elastic water column filling model that introduces the dynamic characteristics of gas can be derived as C + and C － equation.

[0040]

[0041] Furthermore, the step S3 is specifically as follows:

[0042] The simultaneous equations of water and gas total differential equations at the water-gas interface are:

[0043]

[0044] Q w =u g A (22)

[0045] Where Z interface is the height of the air-water interface.

[0046] The present invention provides a simulation method suitable for incorporating gas dynamic characteristics into large-scale water delivery systems to simulate rapid filling processes, addressing the knowledge gap regarding appropriate methods for water filling modeling that incorporate pressurized air characteristics into mathematical model development.

[0047] The simulation results of the model proposed by the invention that takes into account gas properties are compared with the simulation results of the model that considers the correlation between gas pressure change and gas volume change. The former can reflect the difference in pressure changes at different nodes in the gas, while the latter only considers that the pressure in all areas inside the gas is consistent. The comparison of the simulation results of the two models with the experimental data shows that the gas-water transient flow model that takes into account gas properties proposed by the invention more accurately represents the pressure change process of the rapid filling process.

[0048] Importantly, the present invention provides a method for processing the dynamic characteristics of air based on a pressurized elastic water model to establish a rapid water filling process for a large-scale pipeline system. Models that do not consider gas characteristics will underestimate the pressure peak during pressure transients.

[0049] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0050] 1. The gas control equation is incorporated into the pressurized elastic water column filling model, and a numerical model of the rapid filling process considering the gas characteristics is proposed. Due to the representation of the gas continuity equation and momentum equation, the simulation results can show the change of gas pressure at different positions with time, and the gas characteristic parameters density, wave velocity, flow rate, and temperature can all be reflected.

[0051] 2. The quasi-linear hyperbolic partial differential equation governing gas flow was transformed into an ordinary differential equation, facilitating the eigenvalue solution. This transformation reduces solution complexity and simplifies the gas density and velocity as functions of both distance and time. This facilitates analysis of the availability of solutions for the time-varying pressure gradient of the filling water column and yields a specific solution. Therefore, the mature solution to this transformed equation addresses practical problems in transient water-gas two-phase flow during rapid filling processes.

[0052] 3. For the gas region, we applied the derived water-gas two-phase transient flow equation, incorporating the wave propagation and reflection characteristics within the gas, to refine the transient pressure curve due to the cumulative effects of compression and expansion of the air column over time. Furthermore, in cases with higher gas concentrations, the importance of gas necessitates new methods for accurately predicting the transient generation process. An improved numerical model of gas properties, incorporating gas characteristics, has been proven to be a viable approach. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is an implementation flow chart of the present invention;

[0054] Figure 2 A comparison chart of the simulation results of the numerical model incorporating gas characteristics according to the present invention and the original numerical model and experimental results without incorporating gas characteristics;

[0055] Figure 3 These are the simulation results of a large-scale pipeline system (L=2400) using a numerical model that incorporates gas dynamic characteristics and a numerical model that does not incorporate gas characteristics into consideration. Implementation Method

[0056] The present invention is further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, modifications of various equivalent forms of the present invention made by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0057] The present invention provides a simulation method suitable for large-scale water delivery systems that incorporates gas dynamic characteristics, such as Figure 1 As shown, the following steps are included:

[0058] S1: Derive the continuity equation and momentum equation of the gas, and construct the correlation relationship between density, wave velocity, flow rate and other parameters that represent the dynamic characteristics of the gas;

[0059] S2: Use the characteristic line method to transform the gas control quasi-linear hyperbolic partial differential equation into an ordinary differential equation and solve the eigenvalue;

[0060] S3: Solve the entire water-filled pipe by simultaneously deriving the total differential equations of water and gas at the water-gas interface. In this embodiment, the above method is applied as follows:

[0061] In order to verify and analyze the simulation effect of the fast charging process considering the dynamic characteristics of the air mass of the present invention, this embodiment uses the pipeline hydraulic transient experimental device designed by Zhou in 2018 for verification. The system parameters of the experimental device are as follows: pipeline length 8.862 meters, diameter 0.04 meters, upstream water tank head constant 0.08MPa, wave speed 850m / s, initial static state, initial air mass length 0.4 meters, initial air mass pressure 9.794 meters, and the valve opens instantly. The system parameters adopted by the proposed large-scale pipeline system are: pipeline length 2400m, diameter 24m, upstream water tank head constant 0.392MPa, wave speed 850m / s, initial static state, initial air mass length 2390 meters

[0062] The specific implementation steps are as follows:

[0063] S1: Derivation process The continuity and momentum equations of air can be written as:

[0064]

[0065] Where, subscript g is gas; ρ g is the air density; u g is the air speed; P g It is the air pressure.

[0066] The relationship between air wave speed and air density is as follows:

[0067] c g 2 =P g ρ g -1 (3)

[0068]

[0069] Where c g is the air wave speed; m is the constant multidirectional index; c g0 ,P g0 It is c g and P g The initial value of .

[0070] The following equation (5) is derived from the joint solution of equations (3) and (4).

[0071]

[0072] Where k is a constant,

[0073] Substituting Equation (5) into Equations (3) and (4), we can obtain the derived gas continuity and momentum control equations:

[0074]

[0075] S2: Based on the derived continuity and momentum equations of the air, the transformation process of the gas control quasi-linear hyperbolic partial differential equation is carried out to construct the positive and negative gas characteristic equations of the standard one-dimensional unsteady flow in the characteristic line format of the water-filled pipe section:

[0076]

[0077] Where i is the location of the current solution node, t+Δt is the next solution time, D is the pipe diameter, A is the pipe area, and subscript w is water; H w is the pressure measuring head; a w is the wave velocity; G is the acceleration due to gravity; Δx is the unit distance along the pipe.

[0078] The air momentum equation and the continuity equation (6) and equation (7) form a pair of quasi-linear hyperbolic partial differential equations, with the two dependent variables being the velocity u and g and air density c g , the two independent variables are the distance x on the pipeline and the time t. The equation is transformed into an ordinary differential equation using the characteristic line. The simplified air control motion equation and continuity equation can be identified as L1 and L2 respectively from Equations (8) and (9):

[0079]

[0080] These equations are linearly combined with the unknown multiplier λ:

[0081]

[0082] Any two real numbers, with different input values, will again produce two dependent variables u g and c g The two equations are equivalent in all respects. Equations (10) and (11) can be simplified to equation (12) by properly selecting two specific values ​​of λ. In general, the variable u g and c g is a function of x and t, then, from calculus:

[0083]

[0084] Combining equations (11) and (13), we can see that:

[0085]

[0086] Equation (12) becomes an ordinary differential equation:

[0087]

[0088] The solution of Equation (14) yields two special values ​​of λ:

[0089]

[0090] The wave propagation speed c g and speed u g It shows the relationship between the position change of the wave and the time change. When the positive value is taken in equation (16), the positive value of λ must be taken in equation (15). Similar parallelism exists for negative λ. Substituting these values ​​of λ into equation (15) yields two pairs of equations. This pressure elastic water column filling model that introduces the dynamic characteristics of gas can be derived as C + and C － equation.

[0091]

[0092] S3: Combine the constructed gas energy and momentum governing equations and solve them in the characteristic numerical format, and solve the water and gas total differential equations at the water-gas interface to calculate the full-domain water-gas simulation results:

[0093]

[0094] Q w =u g A (22)

[0095] Where Z interface is the height of the air-water interface.

[0096] The derived and simplified governing equations for gas can be combined with the governing equations for water under the characteristic scheme at the boundary to solve the global water-gas transient flow phenomenon. By taking into account the propagation and reflection characteristics of waves within the air column, combined with a numerical model with a mature solution for the transformed equations, it has been proven to be a feasible approach for solving the practical problem of water-gas two-phase transient flow during rapid filling processes.

[0097] Through the programming calculation of the above method, in order to reflect the effect of the method of the present invention, the calculation results of the fast charging process numerical simulation method taking into account the gas characteristics are compared with the experimental results of the embodiment and the original model without considering the gas characteristics. The pressure change curve is as follows Figure 2As shown in the figure, the calculation results of the fast filling process numerical model taking into account the gas characteristics are compared with the original model without considering the gas characteristics in a long pipeline large-scale system. The pressure curve is shown in the figure. Figure 3 shown.

[0098] Depend on Figure 2 It can be seen that compared with the model that does not take gas characteristics into consideration, the fast charging process numerical model provided by the present invention that takes gas characteristics into consideration can accurately describe the transient pipeline pressure amplitude and attenuation.

[0099] Depend on Figure 3 It can be seen that when the pressure results of the fast charging process numerical model provided by the present invention that takes into account gas characteristics are applied to large-scale pipeline systems, the gas characteristics are more obvious, and the original model tends to underestimate the pressure peak when transients occur.

[0100] Therefore, the numerical model calculation incorporating gas characteristics can be well applied to large-scale water transmission systems, solving the problem of accuracy in numerical simulation of water-gas transient flow during fast filling. The proposed scheme can better predict the experimental pressure fluctuations of large air masses and reflect the influence of large air masses on the transient pressure fluctuations of pipelines.

Claims

1. A simulation method for incorporating gas dynamic characteristics into a large-scale water delivery system, characterized in that: The steps include: S1: Derive the continuity equation and momentum equation of the gas, and construct the correlation relationship between the density, wave velocity, and flow rate parameters that represent the dynamic characteristics of the gas; S2: Use the characteristic line method to transform the gas control quasi-linear hyperbolic partial differential equation into an ordinary differential equation and solve the eigenvalue; S3: The water and gas total differential equations are derived simultaneously at the water-gas interface and then solved for the entire water-filled pipe. The gas continuity and momentum control equations in step S1 are derived as follows: The continuity and momentum equations for air are written as: Where, subscript g is gas; ρ g is the air density; u g is the air speed; P g is the air pressure; The relationship between air wave speed and air density is as follows: c g 2 JP g ρ g -1 (3) Where c g is the air wave speed; m is the constant multidirectional index; c g0 ,P g0 It is c g and P g The initial value of The following equation (5) is derived from the joint solution of equations (3) and (4): Where k is a constant, Substituting Equation (5) into Equations (3) and (4), we can obtain the derived gas continuity and momentum control equations:

2. A simulation method for incorporating gas dynamic characteristics into a large-scale water delivery system according to claim 1, characterized in that: The transformation process of the gas-controlled quasi-linear hyperbolic partial differential equation is as follows: for the water-filled pipe section, the positive and negative characteristic lines of the standard one-dimensional unsteady flow are integrated to obtain the following equation: Where i is the location of the current solution node, t+Δt is the next solution time, D is the pipe diameter, A is the pipe area, and subscript w is water; H w is the pressure measuring head; a w is the wave velocity; G is the acceleration due to gravity; Δx is the unit distance along the pipe; The air momentum equation and the continuity equation (6) and equation (7) form a pair of quasi-linear hyperbolic partial differential equations, with the two dependent variables being the velocity u and g and air density c g , the two independent variables are the distance x on the pipeline and the time t. The characteristic line is used to transform the equation into an ordinary differential equation. The simplified gas control motion equation and continuity equation can be identified as L1 and L2 respectively by equations (8) and (9):

3. The method for simulating gas dynamic characteristics in a large-scale water delivery system according to claim 1, characterized in that: Linear combination with unknown multiplier λ: Any two real numbers, with different input values, will again produce two dependent variables u g and c g The two equations represented by are equivalent in all aspects. Equations (10) and (11) can be simplified to equation (12) by appropriately selecting two specific λ values. Generally speaking, the variable u g and c g is a function of x and t, then, from calculus: Combining equations (11) and (13), we can see that: Equation (12) becomes an ordinary differential equation: The solution of Equation (14) yields two special values ​​of λ:

4. The method for simulating gas dynamic characteristics in a large-scale water delivery system according to claim 1, characterized in that: The wave propagation speed c g and speed u g It shows the relationship between the position change of the wave and the time change. When a positive value is taken in equation (16), a positive value λ must be taken in equation (15). Similar parallelism also exists for negative λ. Substituting these values ​​of λ into equation (15) yields two pairs of equations. This pressure elastic water column filling model that introduces the dynamic characteristics of gas can be derived as C + and C － equation:

5. The method for simulating the dynamic characteristics of gas in a large-scale water delivery system according to claim 2, characterized in that: The simultaneous equations of the water and gas total differential equations at the water-gas interface are: Where Z interface is the height of the air-water interface.

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