Modal Analysis and Flow through Sectionalized Pipeline Gate Valve Using FEA and CFD Technique

. Sectionalized gate valves can reduce the volume of product released in the event of a buried pipeline failure or rupture. Te risk of pipeline failure is a constant and common occurrence, and many factors can lead to pipeline incidents. In this paper, the free undamped vibration of the pipeline, sectionalized gate valve structure, and the dynamics of the fuid passing through the system are investigated. First and foremost, a modal analysis based on fnite element analysis (FEA) is introduced as a fundamental linear dynamics analysis to provide insight into how a pipeline sectionalized gate valve structure may respond to diferent types of dynamic loading. Secondly, an implicit numerical analysis using computational fuid dynamics (CFD) is employed to describe physical quantities such as the fow velocity profles at diferent stream positions and pressure felds at diferent points in a control volume. Trough modal analysis, the efective mass factor shows the mass involved in each mode and helps identify modes with high potential to cause damage and prioritize eforts to address them. Te CFD suggests that the sectionalized design of the gate valve leads to a strong vorticity of the fuid in the transversal direction of the fow and a decrease in efciency due to pressure drop.


Introduction
Modern engineering practices emphasize several key factors when designing complex mechanical, aeronautical, or civil structures. Structures must maintain ever-increasing levels of lightness in order to improve efciency and performance. Flexibility must also be incorporated to accommodate a wide variety of operational requirements. Furthermore, the design process must incorporate comprehensive studies and analyses, ensuring the structures meet the necessary standards. For example, automobile manufacturing industries are constantly investing in resources to reduce the weight of their products microscopically. It becomes progressively essential to build engineering structures by adequately considering the dynamics as we better understand them. Engineers gained a versatile conception tool using the fnite element method as a computer modelling approach, particularly for examining dynamic properties. In addition to the dynamic analysis of fnite element methods, this type of numerical analysis requires a rigorous theoretical orientation. Te modal analysis examines the intrinsic dynamics of a system through its natural frequencies, amortization factors, and mode shapes and incorporates this information into a mathematical model to explain its behaviour. Modal model refers to the mathematical formulation of the system, and modal data refer to the information about its features. Physically, a structure's dynamics are decomposed into frequency and position. In the case of continuous systems such as beams and ropes, it is evident from the analytical solution of the equations for the partial derivatives. An invariant linear dynamic system in time can be captured through a modal analysis as a linear combination of simple harmonic movements based on the natural vibrations of the system. Yang et al. studied the process of closing and attenuating the noise of the water hammer caused by the displacement of the valve head, but not the opposite of the outfow of the fuid [1]. During this time, Lai et al. examined the mechanism of anti-check valve opening and proposed a function of approximation showing disk rotational features. In pipeline systems, relief valves are used to control maximum pressure fuctuations [2]. Te CFD technique was used by Zhang et al. to analyze the purgative process and relief valve functionality and internal fow feld of blowassisted pipe [3][4][5]. Tey concluded that the geometry of the relief valve is crucial to its performance, such as the coaxially of the disk. Pipeline codes have complied with a selection of design factors, pipeline location, and maximum distance requirements between sectionalized gate valves [6]. Switching valves enable the piping systems to be turned on and of. According to [7,8], the operation of the drive component in isolation valves for main water and feed water was investigated. In a co-simulation using Adams, UG, and MATLAB, the best pressure for motion performance was determined by developing a dimensionless objective function. Additionally, modal and structural analyses and performance assessment are important components of valve design. Vibration characteristics of the valves are assessed through modal analysis, which is achieved either numerically, experimentally, or theoretically. According to [9], the fapper nozzle valve's modal characteristics are determined numerically by superimposing its natural frequency on top of the fapper armature assembly and its squeal noise was caused by oil pressure pulsation. Zhang et al. investigated the relationship between the mass of the upper platform and the hydraulic stifness of the leg with the electrohydraulic Stewart's sensitivity using natural frequency and mode shape analysis [10]. Lin et al. conducted CFD-DEM simulations to analyze the erosion properties of the gate valve. Te simulations revealed that as the valve opening decreases, the number of particles distributed upstream of the fashboard increases [11,12]. Modal analysis of beam structures has been performed in [13] using multiple-scale random feld models. Tey modelled the heterogeneous material properties and their cross-correlation using random feld theory. Flow resistance characteristics and internal fow characteristics of gate valves are afected by diferent inlet velocity in medium-low pressure gas transmission. A valve's core is often the site of pressure energy loss during gas transmission because the pressure feld distribution, velocity distribution, and velocity streamline distribution depend on the valve opening degree [14].
Sectionalized gate valves enable a reduction of the volume of liquid fowing through the pipeline system. As far as the pipeline and disconnector valves are concerned, the fuid in the pipeline is the only part that is considered part of the system as a result of the pump's action. Based on the modifed Bernoulli equation for incompressible fuid and the polynomial regression function, the volumetric fow rate at the operating point was determined and incorporated as an input parameter into the CFD model to predict the behaviour of the fuid passing through a sectionalized gate valve incorporated into the pipeline system.

Ideal Sectionalized Gate Valve Pipe System
Te sectionalized gate valve pipe system model shown in Figure 1 is a visual representation of a two-dimensional pipe system that shows the components and their connectivity. It is an important tool for understanding the functionality of the system. Te model typically includes the gate valve, pipes, and any other relevant components, arranged in a manner that clearly illustrates their connections and relationships. Tis type of simplifed model is essential in demonstrating the fundamental principles of a gate valve pipe system, making it an important step in the design and implementation of these systems.
Te two-dimensional model in Figure 2 shows a gate valve that can be controlled by handwheels to open or close. Te fuid can fow freely through the valve when the gate or disc is open, allowing it to be elevated out of the fow path. A closed position obstructs the fow of the fuid by lowering the gate or disc. A parallel seat is located on either side of the gate or the disc to guide the disc. A gate valve can only operate in two positions: fully open and fully closed, and it does not provide throttling functionality. Fully open valves reduce energy loss and improve system fow efciency because of the minimal pressure drop across the valve. Figure 1 illustrates a simplifed model of a pipe system that includes a reservoir open to the atmosphere, a pipeline with a sectionalized gate valve, and a centrifugal pump. Fluid fow is controlled by adjusting the valve position on the gate valve in the pipeline. A centrifugal pump increases the pressure of the fuid and transports it through the pipeline from the reservoir to the centrifugal pump. Fluid fow through the pipeline can be controlled with the sectionalized gate valve. Regulating the fow of fuid can be accomplished by closing or opening the valve. An efcient way to control and manage the fow of fuid through a pipeline is by combining a reservoir, centrifugal pump, and a sectionalized gate valve. Table 1 and equation (1) refer to the process of using polynomial regression to model the characteristic curve of a pump. Te manufacturer has provided data points, which are used to ft a polynomial equation. Te order of the polynomial can be changed to produce the best ft for the data (nth-order). Pump head is the dependent variable in this situation, while the volume fow rate is the independent variable. As a consequence of this procedure, the pump performance for various fow rates can be projected using an interpolated pump head function. Te polynomial regression method is an eigen-curve tool for predicting data trends and patterns. However, it can be prone to overftting if the polynomial degree is too high. Additionally, it is sensitive to outliers in the data. Terefore, it is important to check and ensure that the model is suitable for the data and it is providing accurate results. According to least-squares, this function returns a polynomial of the nth order of the volume fow rate Q and the head H that corresponds to the best data ft. It is possible to extract the coefcients of the polynomial regression function from the output polynomial regression function using the method stated in [15].

General Equation Governing the Pipeline System
where a 0 , a 1 , a 2 , and a 3 are third-order polynomial coefcients. Te coefcient number is represented by a constant a 0 without a variable multiplied by it. Polynomial terms characterize each product as denoted by a i Q i . A polynomial function reaches its maximum power at the maximum power of its variable. In mathematics, a leading coefcient is a power term with the highest power. Equation (2) represents the Bernoulli equation modifed to account for the fow of incompressible fuid between a reservoir's surface and its outlet in meters. Te conservation of energy underpinning Bernoulli's principle is explained in the following manner: Pumps transferred a total pressure head to a fuid using a static pressure head P/ρg, a dynamic head v 2 /2g, and a potential head Z above a certain reference level. By summarizing all losses Σh L experienced by a particle, if it fows from point 1 to point 2, the particle's losses (losses ranging from minor to major) are calculated.
Te Darcy-Weisbach equation (equation (3)) is a common method used to calculate head loss due to friction. It relates the head loss to the fuid properties, pipe characteristics, and the fow rate. Based on this equation, engineers can predict the head loss and estimate the pressure necessary to maintain a desired fow rate in the presence of a variety of friction factors, such as the roughness of the pipe, its diameter, and its fow velocity. According to the equation below, the frictional loss, static elevation, and pump head loss are added up to form the total frictional loss: A completely turbulent fow is characterized by a constant friction factor (f ) despite moderate changes in fow. Te friction factor is a dimensionless value that describes the resistance to fow caused by internal roughness in the pipe or duct. Te coefcient of minor loss (K) is another dimensionless value that describes the head loss due to changes in the fow path. In a completely turbulent fow, the head loss due to friction is typically modelled as a proportional function of the fow rate squared, which means that the head loss is directly proportional to the square of the fow rate. Tis relationship is known as the law of fully developed turbulent fow and is represented by the Darcy-Weisbach equation.
Based on pipe diameter, fow velocity, and fuid viscosity, Reynolds numbers (R e ) provide fow state information in the pipeline system on an immediate basis. When the fuid laminar fow is slowly moving, viscous forces dominate, and it appears that the layers are sliding over one another. In a turbulent fow, the fuid particles move chaotically and unpredictably and the inertial forces dominate the viscous forces. It is important to note that the fully turbulent fow is valid only when the Reynolds number is greater than 4000. At this point, the pipe is considered in a turbulent state, meaning that the fow in the pipe is turbulent and the friction factor is constant and independent of the fow rate. If the Reynolds number is below 4000, the fow is considered laminar and the friction factor depends on the fow rate. In the transition zone between laminar and turbulent fow, the Reynolds number is between 2300 and 4000, and the head loss can be modelled using the power-law method.
where Q is denoted as the pipe volume fow rate (m 3 /s), ρ is the density of water at 20°C (989.68 kg/m 3 ), d represents the inside pipe diameter (40 mm), and µ is the fuid absolute viscosity of 1.005 × 10 -3 Ns/m 2 .
When the pipe materials vary in roughness ε of 5.004 × 10 −6 m, consider how friction losses occur as a result of the fuid moving through the pipe.
In Figure 3, the pump characteristic curve is shown against the system curve, which represents the head loss in the system. Te intersection points between the two curves, known as the operating point, refect the volume and the head at which the pump is operating. An efciency of 87.06% can be achieved with a fow rate of 0.0095 m 3 /s, a static elevation of 10 m, and a head of 58.25 m. Tese data can be used to analyze pump performance at diferent fow rates and determine the optimal pump operating point. It is also possible to calculate the pump efciency by comparing the head increase at the operating point with the total load increase at the point on the pump curve when the pump is most efcient. By analyzing this information, faws in pump design or operation can be found.

Finite Element Modal Analysis of Pipeline Sectionalized Gate Valve
Te modal analysis is a useful tool for understanding the dynamic response of a structure to diferent types of loading, such as fuid hydrodynamic forces. In the case of a pipeline sectionalized gate valve, this analysis can help to identify potential resonance vibrations along the system and inform the design to avoid these issues. Te analysis involves determining the natural frequencies and normal modes of the structure, which can be used to predict how the system will respond to diferent types of loading. Te general linear equation governing the motion of the pipeline sectionalized gate valve system can be written by ignoring the damping and external excitation and can be written as where M represents the mass and K represents stifness. Assuming the displacement varies harmonically with respect to time, the displacement u and acceleration € u factors can be expressed as where ϕ i is the modal displacement vector, and substituting (7) and (8) into (6) yields Terefore, the natural frequency f i can be calculated as Te participation factor and efective mass give an idea of how the pipeline sectionalized gate valve structure responds to diferent types of dynamic loading. Participation factor measuring the amount of mass moving in X, Y, and Z directions for each mode is given as    Table 2 shows the relevant parameters of material properties applied in the fnite element analysis (FEA). Inside pipe diameter, the wall thickness, and total length of the model pipe section are 350 mm, 10 mm, and 3.471 m, respectively. Due to the small wall thickness to diameter and axial length of the pipe, linear elastic isotropic materials are the most efective. Finite element analysis is performed using Solidworks. Te fnite element simplifed model of the pipeline sectionalized gate valve structure is shown in Figure 1. Grid size control and pipeline meshing are performed using automatic meshing. Tere are 21280 mesh nodes, 11457 elements, and a maximum aspect ratio of 209.27. Table 3 summarizes the results of natural frequencies and the maximum displacement magnitudes for the diferent mode shapes using Solidworks simulation software. It is noted that the system has fxed constraints located at both ends restricting any movement. Using the maximum displacements of the sectionalized gate valve pipe structure, vibration modes relevant to the fve mode shapes were identifed. Te colour code illustrates the deformation level, for example, the blue colour code illustrates the minimum of the deformed level displaced. Deformation levels were average for the region surrounding the green and yellow colour codes, while they were greater for the region surrounding the orange and red colour codes. Table 4 shows the results of mass participation (normalized) obtained with fnite element modal analysis. Mass participation ratios are important in determining the adequacy of deformation modes to solve dynamic problems with base motion. It is noted that the Y direction, transversal along the pipeline has about 0.82273 normalized mass participating in linear vibration modes. Many codes require that at least 80% of the mass of the system should participate in specifed directions [16].
Te results presented in Figure 4(a) show the displacement amplitude and corresponding frequencies of the valve structure vibration modes. Tis information is important in understanding the dynamic behaviour of the valve and identifying potential issues that could lead to failure. Te displacement amplitude represents the magnitude of the vibration in a given direction, while the frequency represents the number of oscillations that occur in a given time. It is important to note that any external force with a frequency that matches one of the modes of valve structure vibration can cause the valve to vibrate at that frequency. Tis can lead to increased wear and tear on the valve and potentially cause it to fail over time. To avoid this, any external forces with frequencies that match the modes of valve structure vibration must be avoided to keep the valve vibrating under perfect conditions for a long time. Te efective mass factor, as measured in Figure 4(b) for each mode, represents the amount of mass involved in that mode in a given direction of excitation. Tese values are expressed as a percentage of the total system mass and indicate how much of the total mass is involved in a particular mode. Tis information can be used to identify which mode has the most potential to cause damage to the valve and prioritize eforts to address these issues.

Computational Fluid Dynamics (CFD)
Fluid dynamics describe fuid motion mathematically using quantities that pertain to the fuid which are the fow pressure p, fuid viscosity ], fuid density ρ, and fow velocity u. Tere are diferent fow velocities or fow pressures in a control volume at diferent points. Simulation of fuid fow is one of the most important tools used in fuid dynamics. It allows for the evaluation of variations in fow velocity and pressure at diferent points within a fuid domain. Te Navier-Stokes equations govern the motion of the most common liquids, even though all fuids are incompressible to some extent: According to equation (12), the combined efects of advection u∇u and friction can determine the acceleration zu/zt of fuid, particle difusion v∇ 2 u, pressure gradient ∇p/ρ, and body force f. Solidworks fow simulation was the tool used in CFD simulation to evaluate the fow velocity and pressure feld inside the control volume. It was used to execute a simulation of fuid fow behaviour in a sectionalized gate valve pipe system detailing its dynamics.
In inlet boundary conditions, the distribution of variables needs to be specifed in the fow domain at the inlet boundaries, mainly the fow velocity [17]. Table 5 shows the constraints necessary to solve the system of diferential equations governing the fuid dynamics boundary value problem in which the conditions on one extreme of the interval are specifed mostly where the inlet mass fow rate is known. Flow medium is water with a density of 997.56 kg/m 3 at standard sea level. Te results presented in Figure 5 provide insight into the fow behaviour of a fuid as it passes through a sectionalized gate valve. Te fgure illustrates the fow trajectory streamlines, which show the path that the fuid takes as it fows through the valve. Te streamlines can be used to visualize the fow patterns and identify areas of high and low velocity, as well as regions of recirculation and vortex formation. Te inlet mass fow rate of 445 kg/s is applied as the initial fow condition in the control volume in the X global coordinate system. Tis fow rate is considered a turbulent fow longitudinal to the pipe layer. Te turbulent fow can be characterized by chaotic and unpredictable fuctuations in velocity, pressure, and temperature. It is usually found in systems with high Reynolds numbers, where the fuid is highly turbulent and the fow patterns are complex.
Due to the limited computation efort, the selected mesh dimensions are presented in Table 6. Te selected necessary goals of physical parameters that have a substantial impact on the acceptable accuracy of the desired fow without violating the conservation of mass and momentum are shown in Table 7. Figure 6(a) depicts the X-direction fow velocity profle as the fuid moves from upstream to downstream of a sectionalized gate valve. Te velocity in the inlet pipe shows a small    linear increase, but at the sectionalized gate valve, there is an instantaneous fuctuation due to the enlargement of the internal valve area. However, downstream of the valve, there is an exponential increase in velocity which could be caused by the local rotation of the fuid, known as vorticity, between the valve and pipe. It is clearly noted that the amount of mass that passes through the valve is proportional to the fow velocity along the streamline. In contrast, Figure 6(b) shows the Ydirection fow velocity feld, which disappears at the inlet section of the valve. As the fuid passes through the valve, there is a signifcant fuctuation in the nonlinear behaviour of the fow velocity in the Y direction, with small variations downstream of the sectionalized gate valve. Te results shown in Figures 7(a) and 7(b) provide insight into the rotation of the fuid fow through a sectionalized pipeline gate valve. As stated in the fgure's caption, the vorticity along the X and Y directions of the fow is being measured. Te result in Figure 7(a) indicates that in the longitudinal fow direction (X), vorticity is generated by the change in fow direction caused by the valve's sectionalized design. Tis means that the design of the valve causes the fuid to rotate or swirl. A higher impact of vorticity in the Y direction of the fow, as shown in Figure 7(b), suggests that there is a stronger rotation or swirling of the fuid in this direction compared to other directions. Tis can be caused by various factors such as the presence of vortices, turbulence, or shear in the fow. Tese factors can alter the fow patterns and cause pressure drop across the valve, which can impact the efciency of the pipeline system. It is worth noting that the vorticity values in Figures 7(a) and 7(b) are relative to the system and may not have an absolute value that can be compared with other systems. Terefore, the results should be interpreted in the context of the specifc pipeline system being studied.   Figure 8(a) specifcally shows the radial velocity of the fuid, which is a measure of how fast the fuid is moving in a radial direction relative to the valve. Te graph shows that there is a symmetrical instantaneous change when the fow reaches the valve centerline. Tis means that at the centerline, the fuid's radial velocity abruptly changes from a negative value to a positive value. Figure 8(b) shows the circumferential velocity of the fuid, which is a measure of how fast the fuid is moving in a circular direction around the valve. Troughout the internal section of the gate valve, the map shows the distribution of circumferential velocity. It is also discussed in a comprehensive and generalized way how vorticity is transferred as a measure of fuid rotation. Furthermore, the study observes a transient behaviour in the circumferential velocity when the fuid fow reaches the valve. Looking at the inlet section side of the valve, it can be seen that the circumferential velocity is zero until the fow reaches the valve. Te velocity of the fuid changes as it passes through the valve as a result of transient behaviour. In addition, the study notes that attenuation at the outlet section of the valve results in a decrease in fuid velocity as it leaves the valve. Journal of Engineering Figure 9(a) displays three-dimensional (3D) circumferential velocity and longitudinal velocity over a pipe length. A fuid fows circumferentially around the pipe, while a fuid fows longitudinally along the pipe. During a circumferential fow, two diferent speed components are measured: circumferential and longitudinal velocity. According to the fgure, the circumferential velocity at the valve has a lower infuence than the longitudinal velocity. It is observed, however, that both velocities exhibit transient behaviour near the limit of the valve. Accordingly, the longitudinal and circumferential velocities of the fuid change abruptly as it passes through the valve. According to Figure 9(b), the relative pressure fuctuations of the pipe stream are shown in the 3D map. A pressure fuctuation measurement is used to measure how the fuid pressure changes as it fows through a pipe. In comparison to the rest of the pipe, the valve shows a higher and more nonlinear pressure fuctuation. Tere is a sudden change in fow direction and velocity at the valve that can cause this phenomenon. Moreover, it noted that the pressure fuctuation decreases as the fuid passes through the sectionalized gate valve and exits, indicating considerable attenuation downstream.

Conclusion
Te combination of mathematical techniques and numerical analysis was used to study the performance and behaviour of a sectionalized gate valve pipeline system. Specifcally, we used Bernoulli's principle and a third-order polynomial regression with a linear iteration algorithm to determine the characteristics of the centrifugal pump system at its operating point. Te system was solved numerically in a systematic way, taking into account even minor losses in addition to friction losses. Furthermore, the study used FEA modal analysis to determine how the system deforms under external forces. To prevent increased wear and failure, the study emphasizes the importance of avoiding external forces with matching frequencies. For each mode, the efective mass factor shows how much total system mass goes into each mode, and it can be used to identify the modes that pose the biggest risk to the valve. Furthermore, CFD techniques were used in order to analyze the fuid fow through a sectionalized gate valve. It was found that the design of the valve caused the fuid to rotate and swirl, with a stronger rotation in the transversal direction of the fow. Tis can result in a pressure drop across the valve and impact the efciency of the pipeline system. It was also observed that both the circumferential and longitudinal velocities have transient behaviour at the limit of the valve, and the pressure fuctuation is higher and more nonlinear at the valve compared to other parts of the pipe. Some recommendations for future research include the use of other mathematical and numerical techniques to analyze the system's performance and behaviour, such as neural networks, machine learning, and genetic algorithms. Te study of the efect of diferent types of fuids and their properties on the overall system can also be performed.

Data Availability
Data used for the fndings of the study are available on request from the corresponding author.

Conflicts of Interest
Te authors declare that they have no conficts of interest.