Analysis of disc movement during the closing process of axial flow check valves

In a long-distance pipeline system, when the pump is stopped, the axial flow check valve will suddenly close under the pressure difference force of the medium, and the instantaneous valve disc will collide and impact the valve seat at a certain speed, which will have a certain impact on the structural strength and service life of the check valve. This article uses explicit dynamics to simulate the movement process of the instantaneous valve disc when it is completely closed and analyzes the velocity of the valve disc after the collision. Based on the spring damping model theory, the velocity of the valve disc after collision is derived. After error analysis, the reliability of numerical simulation has been verified, which has a certain guiding significance for the practical application and design improvement of axial flow check valves.


Introduction
Axial flow check valves are widely used in long-distance pipeline systems in large-scale petroleum refining, ethylene, and other petrochemical projects, as well as in pipeline systems with harsh operating conditions in large devices, due to their advantages of small flow resistance, stable operation, low noise, and small overall dimensions [1] .Axial flow check valves are key equipment in pipeline systems, whose performance, service life, and operational reliability can all affect the safety of the pipeline system.To determine the safety and reliability of check valves under various working conditions, a large number of experiments are required to verify.With the development of technology, numerical simulation technology has become a reliable method.For complex collision and impact processes, finite element software based on dynamic nonlinear theory is used and this process can be well analyzed and reproduced [2] .
Han et al. [3] selected the RNG turbulence model and used CFD to dynamically simulate and analyze the opening process of a new type of reverse thrust check valve.It was found that the valve disc velocity was close to zero at the moment of complete opening and closing of the valve, indicating that the reverse thrust check valve would not experience impact.Zhang and Yu [4] used numerical analysis software Fluent and dynamic grid technology to analyze the oscillation phenomenon and reasons during the opening process of axial flow check valves, and studied the dynamic characteristics of check valve opening through structural improvement.To extend the service life of swing check valves and reduce energy loss, Zhang et al. [5] established a valve disc motion model combined with a structural dynamics model and used Fluent and Ansys/Ls-Dyna finite element software and analysis modules to simulate and analyze the valve disc motion and impact collision phenomenon during the valve closing process.Zhang et al. [6] obtained the variation law of the inlet medium pressure of the axial flow check valve from the full characteristic curve of the centrifugal pump and used CFD software simulation and UDF dynamic grid technology to dynamically simulate the two-dimensional flow field during the closing process of the axial flow check valve.Due to its short action time and high destructiveness, collisions are difficult for researchers to directly measure.In this paper, the explicit dynamic numerical simulation method is used to simulate the collision and impact process between the valve disc and valve seat when the axial flow check valve is completely closed and the pump suddenly stops.Based on the spring damping model theory, the velocity of the valve disc after collision is derived.After error analysis, the reliability of the numerical simulation is proved.

Basic concepts of impact motion
Collision is a special form of object motion, with a short time history but an extremely complex change process.The collision mechanism is still a hot topic in the theoretical field and many application fields.Due to the large collision force and acceleration generated by object collisions but limited velocity changes, the collision process is usually described macroscopically based on the difference in collision impulse and momentum.
The research in this article belongs to the problem of system dynamic response.For such problems, the dynamic response results of the system after a collision are related to the motion state of the system before the collision, the recovery coefficient during the collision contact process, the collision impulse, etc. [7] .

Explicit dynamics
ANSYS has invested a lot of effort in explicit dynamics, with three explicit dynamics product modules, namely ANSYS LS-DYNA, ANSYS AUTODYN, and ANSYS Explicit Dynamics.LS-DYNA is a world-renowned explicit nonlinear dynamic analysis program that focuses on Lagrangian algorithm, display solving, structural solving, and nonlinear dynamic analysis.It has excellent parallel computing capabilities and excellent mesh adaptation technology [8] .

Establishment of a numerical model
The inlet and outlet diameters of the valve body are both 252 mm, the total length of the check valve chamber is 622 mm, and the front and rear lengths of the valve seat are 156 mm and 462 mm, respectively.The valve disc mass is 4.08 kg, and the spring stiffness is 3,500 N/m.During the working process, the spring is always in a compressed state.When the valve is fully opened, the spring reaches a maximum compression of 86.8 mm, and when the valve is fully closed, the spring reaches a minimum compression of 18 mm.The medium is water.Under normal operating conditions, the medium enters from the left and exits from the right.When the pump is stopped, the medium flows back, and the highpressure medium flows in from the right side and out from the left side.
We grid the model in ANSYS Workbench, and the explicit dynamic analysis diagram and grid diagram are shown in Figure 1.

Analysis settings
We create a 3D geometric model by using SolidWorks and save it.x_t format file is imported into ANSYS Workbench for analysis and use.In engineering data, the performance parameters of the material were input, and to reduce the computational complexity of iterative solutions, the impact geometry model was simplified.The geometric model only contains two components, the valve body and the valve disc.The valve body material is WCB, the valve disc material is CF8M, the number of nodes is 22,322, and the number of elements is 93,480.Given an initial velocity of -3.624 m/s in the initial condition, we keep the remaining settings as default, apply force to the valve body, calculate the k file, and import the k file into the LS-DYNA solver for solving.As shown in Figure 2, at 0.5 ms, the collision contact between the valve disc and the valve seat is completed.Due to the collision effect and energy loss during the collision process, the velocity of the valve disc changes from the initial velocity of -3.624 m/s (direction to the left) to 3.11 m/s (direction to the right).After 0.5 ms, the velocity remains at 3.11 m/s without any change, so the collision time is 0.5 ms.

Establishment of a collision mathematical model
When the pump is stopped, the impact of the valve disc on the valve seat can be simplified as a springdamping model of a mass impacting the elastic wall.The valve disc accelerating along the negative x direction eventually collides with the valve seat at a speed of 3.624 m/s, corresponding to the free mass colliding with the elastic wall at a positive speed, as shown in Figure 3.The stiffness coefficient and damping coefficient of the elastic wall are denoted as k and c, respectively.When t=0, it is the start time of the collision.During the collision period, the motion of the free mass satisfies the following initial value problem [9][10] .
During a collision like this, the motion of a moving object that undergoes repeated vibrations to reach its equilibrium position belongs to underdamped motion.For the underdamped case, the solution of the differential equation (i.e., Equation ( 1)) is: v is the initial velocity of mass motion, m is the mass of a freely moving mass, and t is impact time.

Optimization of k and c in Maple
In practical engineering applications, it is difficult to determine the actual values of spring stiffness k and damping coefficient c, so it is necessary to obtain the values of k and c under the research conditions.
The following is to optimize the accurate values of k and c in Maple.
After transforming the above parameters , , d A   into functions related to m, k, and c, it can be concluded that: Substituting Equations ( 3), (4), and (5) into Equation (2) yields: We write a Maple operation program according to Equation (6).The relationship between the maximum impact displacement ( ) x t , damping c, and spring stiffness k obtained in Maple is shown in Figure 4.As the damping coefficient increases, the maximum impact displacement gradually decreases in a parabolic shape.As the spring stiffness increases, the maximum impact displacement decreases in a straight line. .Based on the curvature radius of the contact circle 104mm  r , the maximum impact displacement along the radius r direction of the vertical direct contact surface is obtained as x t x x r r . Under this condition, the two-dimensional relationship between k and c is shown in Figure 6.As the damping coefficient continues to increase, the spring stiffness roughly increases in a straight line with a slope of 1.  G  MPa, and thus G=8, 200 MPa.Considering the collision system as a cube with side length, the spring stiffness is obtained as 8 5  5 82000 622 2.55 10 N/mm       k Ga [11] .When the spring stiffness is

Theoretical derivation process
Taking the first derivative of Equation (2) yields: , ( ) exp( )( cos sin ) Taking the second derivative yields: ,, We order t t   and ,, ( ) 0 x t  and then obtain: The contact time is the minimum positive solution of Equation ( 8), which is: The velocity of a free mass before collision is known to be 0 v .Through the relationship between velocity before and after collision, we obtain the velocity of the free mass at the end of the collision We substitute the initial conditions 0 into Equation (10) to obtain the final post-collision velocity as 1 2.985m/s v  .The equation for Newton's recovery coefficient is: We substitute 1 v into Equation ( 11) to obtain Newton's recovery coefficient of 0.825.

Error analysis
The errors generated above belong to model errors and truncation errors, which are generated during theoretical derivation and simplification of the model.In numerical simulation calculations, discretization of continuity equations results in truncation errors.

Conclusion
1) The velocity of the valve disc after collision is obtained through numerical simulation of display dynamics at 3.11 m/s; 2) In theoretical derivation, due to the difficulty in determining the damping coefficient c in the spring damping model, the impact dynamics equation was used to optimize the relationship curve between k and c in Maple, in which and c=1, 400. 3) According to theoretical derivation, the velocity of the valve disc after collision is 2.985 m/s, and the relative error compared with the numerical simulation value is 4.2%, proving the reliability of the numerical simulation.
In summary, the research in this article has certain guiding significance for the practical application of axial flow check valves under pump shutdown conditions.

Figure 1 .
Figure 1.Explicit dynamic analysis diagram and grid diagram.

Figure 2 .
Figure 2. The change diagram of disc velocity at the moment of collision.As shown in Figure2, at 0.5 ms, the collision contact between the valve disc and the valve seat is completed.Due to the collision effect and energy loss during the collision process, the velocity of the valve disc changes from the initial velocity of -3.624 m/s (direction to the left) to 3.11 m/s (direction to the right).After 0.5 ms, the velocity remains at 3.11 m/s without any change, so the collision time is 0.5 ms.

Figure 3 .
Figure 3. Collision between a free mass and an elastic wall.

Figure 4 . 3 2
Figure 4.The three-dimensional relationship ( ) x t of k and c.

Figure 5 .
Figure 5. Collision diagram.Figure 6. Relationship curve of k and c.The calculation equation for spring stiffness is 5 k Ga  , in which G is the shear elastic modulus.The shear elastic modulus of the valve body WCB is 1 84000Mpa  G .The shear elastic modulus of the valve disc material CF8M is 2 80000G  MPa, and thus G=8, 200 MPa.Considering the collision system as a cube with side length, the spring stiffness is obtained as

Figure 6 .
Figure 5. Collision diagram.Figure 6. Relationship curve of k and c.The calculation equation for spring stiffness is 5 k Ga  , in which G is the shear elastic modulus.The shear elastic modulus of the valve body WCB is 1 84000Mpa  G .The shear elastic modulus of the valve disc material CF8M is 2 80000G  MPa, and thus G=8, 200 MPa.Considering the collision system as a cube with side length, the spring stiffness is obtained as According to transient dynamic numerical simulation, the velocity of the valve disc after contact collision is 3.11 m/s, the contact collision velocity obtained from theoretical derivation is 2.985 m/s, and the relative error obtained is: