Introduction to the equation of water hammer motion in infusion pipeline

Now the main research is to set up an air tank on the infusion pipeline to effectively control the water hammer, the basic differential equations of unstable flow, composed of 2 differential equations: equation of motion: 9v9t v9v9x 1Q9p9x gsinH K2Dv | v | = 0 (1), Continuous equation: 9p9t v9p9x Qa29v9x = 0 (2), where: t is time; v is flow rate; Q is liquid density; p is hydraulic pressure; a is water hammer wave propagation speed; g is free fall acceleration; x is distance; H is the horizontal inclination of the pipeline; K is the resistance factor of Darcy; D is the inner diameter of the pipeline.

V9v9x in equation (1) is very small compared to 9v9t and can be ignored; and because the influence of gravity is not included in the steady-state continuous equation, it should also be ignored in the transient state, so v9p9x in equation (2) can also be ignore. According to the characteristic line method, the two partial differential equations are converted, and the height H of the hydraulic slope line from the reference plane is used to replace the dynamic water pressure p, the flow rate Q is used to replace the flow velocity v, the cross-sectional area of ​​the pipeline is X, and the friction resistance is written as In the form of Repinzon's formula, the following approximate water hammer characteristic equation can be obtained: C dxdt = a (3), agXdQ dH fQ | Q | 1-mdx = 0 (4), C-dxdt = -a (5), agXdQ-dH fQ | Q | 1-mdx = 0 (6). Equations (3) and (5) are called characteristic line equations, and equations (4) and (6) are called compatibility equations.

The finite difference method can effectively solve the integral, so the difference method can be used to solve the characteristic equation. From equation (3) and equation (6), the following equations are obtained: C: HP = RA-SAQP (7), C-: HP = RB SBQP (8), where: RA = HA CWQA, SA = CW f | QA | 1-ma $ t; RB = HB-CWQB, SB = CW f | QB | 1-ma $ t; CW = agw; HP, QP are the pressure and flow at the P node of the pipe segment at time t; HA, QA are: The pressure and flow at the node of the pipe section at time t and $ t; HB and QB are the pressure and flow at the node of the pipe section at t- $ t respectively; f is the friction coefficient along the way. The values ​​of HP and QP at the intermediate nodes of the pipeline can be obtained from equations (7) and (8).

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