Numerical simulation of the erosion of pipe bends considering fluid-induced stress and surface scar evolution

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Numerical simulation of the erosion of pipe bends considering fluid-induced stress and surface scar evolution

T

Huakun Wanga,b,∗∗, Yang Yua,b,∗, Jianxing Yua,b, Weipeng Xua,b, Xiubo Lic, Sizhe Yua,d a

State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin, 300072, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, 200240, China c CRRC SMD (SHANGHAI) LTD, Shanghai, 201360, China d Xiamen Port Construction Group Co. LTD, Xiamen, 361006, China b

ARTICLE INFO

ABSTRACT

Keywords: Stress–erosion Pipe bend Fluid–structure interaction Multiphysics

Erosion–corrosion is the main cause of pipeline failures, especially in pipe bends, plugged tees, and pipe fittings with different cross sections. In this study, the erosion of a stressed pipe bend (arising from the internal highpressure fluid) caused by particles entrained in the conveying fluid was investigated based on a numerical analysis. A numerical model based on the authors’ previous work was combined with a new erosion equation that considers the effect of the tensile load on erosion to analyze the erosion mechanism of a pipe bend under high pressure. The numerical model was verified by the erosion test of API X65 pipeline steel, and a systematic parametric study was also carried out. Moreover, a new numerical simulation method that considers the surface scar evolution of a pipe bend based on the erosion history, was proposed. This work provides an alternative means of understanding erosion behavior and predicting erosion damage of in-service, highly stressed oil and gas pipelines.

1. Introduction Pipelines are widely used in both onshore and offshore oil and gas transportation owing to their high efficiency and low cost. Given this wide application, pipe integrity warrants careful study. The literature indicates that mechanical damage, corrosion (external and internal), erosion, construction defects, material failure, natural hazards, structural threats, and operational unpredictability are the most common reasons for pipeline failure [1–15]. For pipelines transporting crude oil or natural gas, the presence of water, oxygen, solid particles, and salts threatens the integrity of the pipeline owing to the erosion–corrosion (E–C) mechanism [10,13–15], which reduces the pipe strength. For onshore pipelines, an E–C defect may lead to leakage and a burst failure mode, whereas for deep-sea pipelines, an E–C defect may lead to leakage and a collapse failure mode [4–6,10]. It has been widely observed that the overall loss in mass owing to the interaction of erosion and corrosion is much higher than the sum of the mass losses owing to each of these components acting separately [13–15]. The results of an impingement jet test on X65 steel in an oil–sand slurry and aluminum

∗

alloy in an ethylene glycol–water solution indicated that erosion dominated the E–C process, accounting for 70–97% of the total E–C rate [16–18]. In practical engineering applications over the past few years, studies have attributed numerous pipe failures to flow accelerated corrosion (FAC) or the E–C mechanism, especially failures in pipe sections subjected to sudden changes in flow direction, such as 90° elbows [19,20]. Over the last few decades, the factors governing E–C have been well documented. Of these factors, particle velocity (up) and impact angle (α) are two key parameters that govern the erosion of a target material [21]. For brittle materials, the erosion rate increases with the increase in impact angle up to 90° [22,23], whereas for ductile materials, the relationship is more complicated. Generally, the maximum erosion rate is observed when the value of α is in the range of 20–45° [24,25]. Moreover, particle properties such as size, shape, hardness, and density also have a great impact on the erosion of target materials [26]. This is particularly true for particle size and shape. To date, few papers have addressed the statistical characterization of the properties of particles and the corresponding numerical simulation methods [27–29], which

Corresponding author. State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China. Corresponding author. State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China. E-mail addresses: [email protected] (H. Wang), [email protected] (Y. Yu).

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https://doi.org/10.1016/j.wear.2019.203043 Received 4 June 2019; Received in revised form 3 September 2019; Accepted 3 September 2019 Available online 05 September 2019 0043-1648/ © 2019 Elsevier B.V. All rights reserved.

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Fig. 1. Model details of pipe bend.

conditions and found the wear rate to be closely related to the water pressure. Zhang [33] studied the slurry erosion of high-pressure pipelines during hydraulic fracturing slurry flow using the failure analysis of high-pressure elbows based on both their macroscopic features and a scanning electron microscope (SEM); some additional tests were also carried out to reveal the erosion mechanism. A novel test setup was developed by Sun et al. [25]; slurry impact erosion tests at different impact angles from 15° to 90° with the impacting surface under different levels of tensile stress (0–500 MPa) were performed, and the dependency of the erosion rate on applied stress was determined. Drawing on Sun's work [25], Wang et al. [21] developed a novel empirical erosion equation, which is a modified version of the E/CRC model that allows for the effect of material stress on erosion, and a numerical simulation method was also proposed for the first time. To date, more than 200 erosion models have been proposed [26,34]. Of these, only 28 models are considered particularly important [35], amongst which the DNV [36], E/CRC [37,38], Oka [39,40], and Finnie [41] erosion equations are the most widely used owing to their relative simplicity and accuracy. All pipelines used for the transportation of oil or natural gas must withstand tensile stress arising from internal pressure or axial tension stress (such as in the case of a marine riser). For ultra-deep-sea pipelines, the pipe wall may additionally suffer from compressive stress. Except for the erosion equation proposed by the authors [21], none of the aforementioned erosion models have considered the operational stress. Moreover, to the best of the authors’ knowledge, no study addressing the erosion prediction of high-pressure pipelines using numerical methods exists. In the current work, drawing on the aforementioned prior research [21], the stress-erosion model from Ref. [21] was used to explore the effect of pipe wall stress on the erosion rate of the pipe bend under real-world in-service conditions, by considering the tensile stress arising from the internal fluid pressure. A fluid–structure interaction (FSI) model was developed, and the evolution of the surface scar was studied based on the erosion history. This approach can also be applied to the prediction of erosion in other high-

Fig. 2. Schematic of the mesh and mesh convergence test (Vin = 4 m/s).

provide the foundation for the investigation of particle erosion. Note that most of the erosion tests in the literature have been performed in small-scale pipes with air and water as the fluids being conveyed under atmospheric pressure; only a small number of tests have been carried out under the high operating pressures that are experienced by in-service pipes used in real-world engineering situations. A study by Evans [30] details the use of natural gases as the fluid with a maximum operating pressure of 6.89 MPa, whereas a study by Kvernvold and Sandberg [31] used water and nitrogen at 7 bar. Wang [32] evaluated the erosion of four types of alloy (316 steel, Hastelloy C-276, Inconel 625, and Ti6Al4V) under simulated deep-sea environment Table 1 Fitting parameters of the user-defined erosion equation with tensile stress [21]. Model parameters E/CRC-Log-S

C′ −7

1.212 × 10

μ1

σ1

a

b1

c1

b2

c2

w

−0.02274

0.8549

−132.7

129.9

129.9

−50.22

0

0.8118

2

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Fig. 3. Flow chart of the simulation process.

Generally, for predicting the erosion of a pipe bend with a dilute phase flow (one-way coupling), three steps have to be carried out [21,43–47]: (1) simulate the flow of the carrier fluid by solving the Reynolds-averaged Navier–Stokes equation, (2) perform particle tracing based on the fluid field, and (3) predict the erosion rate based on the erosion equation and particle impact information. This is the standard procedure for erosion prediction using a traditional erosion model (such as DNV and E/CRC). For stress–erosion prediction, an additional solid mechanics analysis step is required [21]. For highpressure pipelines, as the circumferential tensile stress acting on the pipe wall is arising from the internal high-pressure fluid, the FSI must be considered; more detail on this is provided in Section 2.1.3. Note that any other load acting in the axial direction of pipe, such as that owing to the self-weight or ground motions, was ignored.

Table 2 Parameters of the mesh convergence test. No.

No. No. No. No. No. No. No.

1 2 3 4 5 6 7

Wall thickness direction

Element number every 90° in the circumferential direction

Number of boundary layer

Domain mesh size

Total number of elements

2 2 2 2 2 2 3

8 10 15 10 10 10 10

8 8 8 8 4 12 12

Coarse Coarse Coarse Fine Coarse Coarse Coarse

20938 26854 41702 50244 22214 31494 32654

pressure pipe fittings, such as plugged tees or valves. Note that the contribution of corrosion to the damage of the pipe bend was ignored, which means only the promoting effect of stress on erosion was considered.

2.1.1. Governing equations of fluid motion For fluid flow with a high Reynolds number (Re > 2300 [48]), which is the case in most practical engineering applications, a turbulence model with wall functions is required [49]. In this study, the k–ω model was adopted owing to its robustness in modeling fluid flows involving a strong streamline curvature [21,42,50], and the effect of gravity on fluid flow was also considered. The governing equation in Step (1) has been discussed in detail in the authors’ previous work [21]. The inlet velocity (Vin) of the fluid obeys the 1/7 power law, and a specific outlet pressure (Pout) was applied. A boundary layer mesh was also adopted to accurately capture the motion of particles near the wall. Moreover, a combination of a structured and free triangular mesh was also adopted to reduce the computational cost (see Fig. 2).

2. Numerical simulation method A numerical model in which the FSI is considered, is described in this section. The proposed stress–erosion equation [21] was integrated into the commercial software COMSOL Multiphysics version 5.3a [42] to facilitate the investigation of the tensile loading effect on erosion and erosion evolution. 2.1. Model details

2.1.2. Particle tracing Following Step (1), particle tracing should be performed. The governing equation of particle motion is based on Newton's second law, which can be written as follows [42]:

Stress–erosion is a complex phenomenon in which solid mechanics, hydrodynamics, and particle tracing are involved [21,25]. To improve computational efficiency, a simplified three-dimensional (3D) model (Fig. 1) was established. Although only half the model was established initially, the full model can be visualized during post-processing. Also shown are the schematic diagrams of the computational domain and the boundary conditions. More detailed modeling information is provided in the following sections.

d(mp V dt

= Ft

(1)

where mp is the particle mass, v is the velocity vector of the particle,

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Particle Density ρp (kg/m3)

2650

Particle Diameter dp (μm)

400–500 (450 averaged)

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and Ft is the resultant force acting on the particle. In this study, only the drag force FD, gravitational force Fg, pressure gradient force Fp, and the added mass force Fa were considered. Other forces, such as the Basset force, Magnus force, and Saffman lift force, were ignored. Generally, the turbulent dispersion of particles in a turbulent flow is modeled using an eddy interaction model [51]. The discrete random walk model was adopted in this study. Based on the aforementioned assumptions, each term of the load can be written as follows [42,43,52]:

Ft = FD + Fg + Fa + Fp FD =

mp u

4 p dp2 3µCD Rer

Semi round (Fs = 0.53)

v),

u = u + u,

, CD = f (Rer ),

Rer =

2k 3

u= u

v dp µ

p p

Fa =

1 dv m 2 p dt

Fp =

Vp

p

=

1 6

dp3

(2)

p

Sand

where u is the velocity vector, k is the turbulence kinetic energy, ζ is a vector of uncorrelated Gaussian numbers with unit variance, ρp is the particle density, ρ is the density of the fluid, dp is the particle diameter, Rer is the relative Reynolds number, and Vp is the particle volume. Owing to the multiphysics involved in the stress–erosion process, only a one-way coupling analysis (suitable for dilute phase flows with particle concentrations < 1% wt) was implemented to simplify the simulation process and improve the computational efficiency. Chen [43] conducted flow loop tests (gas-solid) to determine the erosion characteristics of pipe bends and plugged tees, and some impingement jet tests (both gas–solid and liquid–solid flows) were performed by Mansouri [44]. In their numerical simulation, 50000 particles were released from the nozzle inlet to ensure that the averaged impact data are not a function of particle number [44], a concept that was also adopted in this paper. Moreover, Grant and Tabakoff's [53] test-based stochastic particle rebound model was adopted to simulate the particle–wall interaction. In practice, the size distribution of the sand used should be considered [54]; however, no detailed information about the size distribution could be found in Zeng's [54] test results, which were used for model validation. As such, all particle diameters were assumed to be 0.45 mm (mean value) in this work.

5.4 235 60 4 2

=

p

Fg = mp g

2.1.3. Solid mechanics and fluid–structure interaction For safety considerations, the parameters of pipelines (steel grade, diameter, and wall thickness) were selected based on the stress-based design criterion, in which the in-service pipe wall stress is limited to a fraction (≤90%) of the specified minimum yield strength [55]. Because the internal fluid-induced tensile stress in practice was in the elastic range, an isotropic linear elastic material model was adopted. The constitutive model obeys Hooke's law [21,42]. Considering that the structures are in the elastic deformation range, only a one-way FSI coupling was adopted, and thus, only the effect of fluid pressure on the pipe wall was considered; the structural deformations are so small that they do not significantly affect the fluid domain. The FSI couplings appear on the boundaries between the fluid and the solid domain, and the total force exerted on the solid boundary by the fluid is the negative of the reaction force on the fluid [42]. Given the abovementioned assumptions, the governing equation for the fluid–structure interface can be written as follows:

1

Mass Flow Rate of Sand Particles (g/s) Inlet Velocity Vin (m/s) Wall Thickness t (mm)

solid· n fluid

=

=

fluid· n

pfluid I + µ ( ufluid + ( ufluid)T )

2 µ( 3

· u) I

(3)

where p denotes pressure, μ is the dynamic viscosity of the fluid, n is the outward normal to the boundary, and I is the identity matrix.

50

Inner Diameter Di (mm)

Table 3 Zeng's test parameters [44,54].

Outlet Pressure Pout (atm)

Temperature T (°C)

pH

Particle Material

Particle Shape

p

1

4

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Table 4 Material parameters of API X65 steel [54,56]. Material

Young's Modulus, E (GPa)

Poisson's Ratio, ν

Yield Strength, σy (MPa)

Density, ρ (kg/m3)

BH

X65 pipeline carbon steel

206.1

0.3

448

7850

200

Table 5 Distribution of the pure erosion rate at ϕ = 180° (see Ref. [54] Fig. 4 (b)). θ (degrees)

11.50

22.78

33.76

45.17

56.31

67.56

78.70

Pure erosion rate (ug/cm2/h)

58.61

76.49

119.00

153.33

179.38

237.52

282.99

Fig. 4. Results of the mesh convergence test.

2.1.4. Stress–erosion prediction In the authors’ previous work [21], an erosion equation, based on test data of Sun [25] and considering the applied tensile stress, was proposed. It can be written as follows:

ER = C (BH )

F1 ( ) =

0.59F u 2.41 F ( s p 1

1 1 2

F2 ( ) = a +

)exp

mises y

Fig. 6. Comparison of the simulation results produced by alternative erosion models and the test data [54].

· F2 ( )

The corresponding model parameters are listed in Table 1. Because the coupling effect between different types of physics is weak, the sequential coupling method was adopted [21]. As such, the turbulent flow was established first, and then the stress field on the pipe wall was calculated based on the obtained fluid pressure acting on the inner pipe wall. Thereafter, the particle tracing process was completed based on the fluid field. Finally, the erosion damage was obtained by accumulating the predicted ERi given by Eq. (7) based on the calculated von Mises stress (σMises), velocity (upi), and impact angle (αi) of the ith particle. The flow chart of the simulation process is shown in Fig. 3. The

1(ln µ1)2 2 2 e 1 n i=1

[bi sin(iw ) + ci cos(iw )]

(4)

where ER is the erosion ratio with the units “kg/kg” (meaning “kg of surface material removed per kg of particle impacting that surface”), BH is the Brinell hardness of the target material, Fs is the shape factor of the particle, up is the impact speed of the particle (m/s), α is the impact angle (rad), σMises is the von Mises stress, σy is the yield stress, and C′, μ1, σ1, a, b, and c are constants.

Fig. 5. Schematic of the damaged pipe bend model.

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an indicator of mesh convergence. For the mesh convergence test, both the upstream and downstream lengths of the pipe bend were chosen to be 4Di, and the model parameters and simulation results are shown in Table 2–Table 4 and Fig. 4. Based on the predicted results of Nos. 1, 2, and 3, ten elements per 90° of circumference are sufficient. Even though a finer mesh in the axial direction (No. 4, control by domain mesh size) gives a smooth erosion distribution in the extrados of the pipe bend, the predicted maximum erosion rate is basically the same, whereas the computational cost increases sharply; we think such a refinement is unnecessary. The predicted results of No. 2, 5, and 6 indicated that 8 boundary-layer elements are sufficient; however, considering erosion scar, 12 boundary-layer elements were adopted. The predicted results of No. 6 and 7 indicated that 2 elements in the wall thickness direction are adequate. In the following section, the mesh scheme of No. 6 was adopted. For the damaged model (see Section 3), more elements in the pipe bend (18 elements along the axial direction) were used because of the complex geometry. 3. Methods used for the prediction of the evolution of the erosion scar Fig. 7. Comparison of the predicted erosion rate with different lengths L2 (along the axial direction in the symmetry plane).

As particle erosion proceeds, the pipe surface evolves, changing the fluid flow. This in turn changes the particle motion, affecting the erosion of the pipe wall. To take these factors into account, a new method was developed. First, an intact pipe bend was used to simulate the erosion of a high-pressure pipe. The erosion rate (ERa, the unit is kg/ (m2·s), as is often the default reporting unit in CFD packages) on the inner surface of the pipe wall was extracted and then used to construct a damaged-pipe model incorporating an erosion scar. The shape of the erosion scar was obtained by assuming the erosion rate to be constant over a relatively short time interval △T, and the erosion depth dER was

difference between this study and the authors’ previous work [21] is that, here, the load acting on the pipe wall is arising from the highpressure fluid, and the FSI must be considered, whereas the load acting on the plate specimen arises from the tensile load [21]. A mesh convergence study was carried out to obtain a reasonable mesh size to ensure accuracy at the lowest computational cost. The mesh size was refined, and the erosion rate on the pipe wall was used as

Fig. 8. Von Mises stress distribution of the pipe wall with different internal pressures.

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Fig. 9. Effect of internal operation load on erosion. 7

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Fig. 10. Impact angle on the inner surface of the pipe bend and downstream straight section.

Fig. 11. Normalized erosion ratio as a function of the impact angle.

obtained by ERa/ρ·△T (ρ is the density of the target material; for the environmental parameters used in our numerical model, △T is approximately 45 days for the increase of 1 mm of dERmax). Next, the new damaged-pipe model was used to carry out an intact simulation following the procedure shown in Fig. 3. Two factors were considered: the change in particle motion owing to the change in fluid field and the stress acceleration effect enhanced by the stress concentration owing to the existence of the erosion scar further promoting the erosion process. The hotspot (the site with maximum erosion depth, dERmax) of this erosion scar may vary owing to the coupling effect of the particle motion and stress concentration. The numerical model of a damaged pipe was established by lofting 11 cross sections (every 10°, as in Fig. 5) with different erosion depths. The coordinates of the control points for each section were first determined based on the erosion rate and then connected by an interpolation curve. Further details of this model are shown in Fig. 5.

the test data given by Zeng [54] were adopted. In the test of Zeng, the E–C behavior (total E-C rate and pure erosion (erosion with no corrosion present) rate) of an X65 pipeline carbon steel at the elbow was obtained using array electrodes; the test was repeated, and the test data were also well organized. The distribution of the pure erosion rate on the extrados of the pipe bend was extracted and compared with our numerical results. Detailed information on the E–C test is presented in Tables 3 and 4. Following the process outlined in Zeng's study [54], the upstream length of the elbow entrance was set to 1 m to facilitate a fully developed flow and to ensure that a sufficient dispersion of sand particles occurs in the stream before it enters the pipe bend. For the consideration of computational cost, the length of the downstream straight pipe was varied and the predicted erosion rate was compared, which indicated that the predicted results given by 6Di and 3Di are almost identical. Hence, the downstream length of the elbow exit was set as 3Di, which was adopted throughout the study. Based on the geometry parameters and flow conditions, a high Reynolds number (Re) was obtained (Re = ρVd/μ which is approximately equal to 2 × 105 > 2300) [48]. Although no detailed information was provided about the shape of the sand particles used in Zeng's study [54], the observation results obtained via SEM [54] indicated the shape to have

3.1. Model validation The numerical model should be validated before further analysis. Considering that the test data obtained from a flow loop system are more consistent with engineering practice and our numerical model,

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Fig. 12. Effect of fluid velocity on the erosion rate.

been round, semi-round or sharp. Particles in this study were, therefore, assumed to be semi-round in shape. Based on the above information, the erosion rate of the pipe bend was obtained. This was compared with the simulation results obtained via alternative erosion equations (DNV and E/CRC). Table 5 shows the distribution of the pure erosion rate on the extrados of the pipe bend. The erosion depth increases from the elbow inlet to the elbow outlet, with a maximum near the outlet, while no test data is available downstream of the pipe bend. However, the numerical simulation results indicated that the erosion scar then continues into the straight section of the pipe downstream of the elbow. This is discussed in detail in Section 5. As shown in Fig. 6, the predicted results obtained via the user-defined erosion equation and the DNV model are in good agreement with the test data, whereas the E/CRC value obtained was significantly overestimated. One major reason may be that the test conditions used to develop the E/CRC model were based on a series of direct impact tests of Inconel 718. In these tests, sand particles with a mean size of 150 μm are injected into high velocity airflows [37]. However, in Zeng's test, the carrying fluid is water and the particle size is approximately 450 μm, which is consistent with the test conditions of Sun [25], where the carrying fluid is also water and the mean value of the particle size is 550 μm. Moreover, the stresserosion equation was developed by the authors [21] based on the test data of Sun. Additionally, it appears all models significantly underpredict the erosion at the elbow entrance. The reason may be that in the

Fig. 13. Maximum erosion rate as a function of the fluid velocity.

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Fig. 14. Effect of the particle size on the erosion rate, Vin = 4 m/s, Pout = 15 MPa.

real system, the particles have greater turbulent motion than that which is captured by the current modeling, leading to greater erosion at the elbow entrance. The reason for the predicted result obtained via the user-defined stress-erosion equation being almost identical to that without the consideration of applied stress is that the outlet pressure is equal to atmospheric pressure, and the fluid-induced stress is very small. Therefore, in this case, the contribution of stress to erosion is almost negligible. Having established the reliability of the numerical model, it was used to carry out further analyses as described in Section 5. 4. Parametric study and discussions Based on the aforementioned modeling information, the effects of fluid velocity Vin, particle diameter dp, and density ρp, the internal pressure (Pout) and inner surface evolution on erosion were systematically studied. Considering that the numerical model used in Section 4 is extremely computationally expensive, the length upstream of the pipe bend (L2) was varied, and the predicted erosion rate was compared. The simulation results, shown in Fig. 7, indicated that the length of the upstream straight pipe greatly influences the predicted erosion rate; this is attributed to the fact that it takes time for particles deposited under gravity to reach a relatively steady state. The simulation results also indicated that a length of 16Di for the upstream distance is adequate for

Fig. 15. Maximum erosion rate as a function of the particle diameter (ρp = 2650 kg/m3).

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Fig. 16. Effect of the particle size on erosion (Vin = 4 m/s, Pout = 15 MPa, ρp = 2650 kg/m3). (a) dp = 150 μm (b) dp = 250 μm (c) dp = 350 μm (d) dp = 450 μm.

Fig. 17. Effect of the particle density on the erosion rate, Vin = 4 m/s, Pout = 15 MPa.

use in the simplified model in the following section. The slight difference in erosion rates (Fig. 7) is mainly owing to the differences in the motion information of the random wandering particles and the accumulation algorithm used to calculate the erosion rate.

4.1. Effect of operational pressure This section provides a detailed analysis of the use of different operational pressures and the prediction of the erosion rate using the userdefined erosion equation both with and without the consideration of fluid-induced stress. The simulation results obtained using DNV and E/

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also compared; in all cases, the erosion rate predicted using DNV was 50% higher or more. To investigate this phenomenon, the impact angles in the region of the pipe bend and the downstream straight pipe were extracted separately; the statistical results are shown in Fig. 10. It can be clearly seen that the impact angle on the inner surface of the pipe bend (at first impact) ranges from 10° to 20°, while the impact angle on the inner surface of the downstream straight section ranges from 0° to 10°. Based on the normalized ER ratio (defined as the ratio of the predicted ER given by different erosion equations) shown in Fig. 11, it is clear that for a lower impact velocity (< 5 m/s), although no great difference exists between the predicted ER of the user-defined erosion equation and that of the DNV model when the impact angle ranges from 15° to 50°, beyond this range the predicted ER obtained via the userdefined erosion equation is much lower, especially for impact angles less than 15°. The user-defined erosion model was developed based on the test data of Sun [25], while no test data below an impact angle of 15° were available. Owing to the inherent nature of the probability density function (PDF) of the lognormal distribution (which was used as the angle function [21]), the value of the PDF decreases quickly when the impact angle is less than 15°, and the erosion rate may be over underestimated in this case. However, considering that the maximum erosion rate occurs at an impact angle of approximately 30°, such an underestimation can be ignored. The velocity dependency in the figure on the right in Fig. 11 is mainly attributed to the different value of n adopted by the erosion model. The striated distribution of the impact angle shown in Fig. 10 (left-hand image) can be largely attributed to the loft-based modeling technique; although a denser loft section makes the calculated impact angle distribution more uniform, owing to the complexity of the modeling process, this was deemed unnecessary. As the operational pressure increases, the contribution of stress to the erosion process is predicted to become more significant, especially in the extrados in the symmetry plane, where the maximum erosion rate occurs (Fig. 9). The simulation results indicated that the traditional erosion model may underestimate the risk of leakage in high-pressure pipelines, emphasizing the importance of considering the effect of the operating load to effectively monitor the structural health.

Fig. 18. Maximum erosion rate as a function of the particle density (dp = 450 μm).

CRC are also presented. For a straight pipe with a circular cross section, the correlation between the stress (σ) on the wall and the internal pressure (Pi) can be expressed using Eq. (5) in the elastic range:

=

Pi Di 2t

(5)

where Di is the internal diameter of the pipe, and t is the wall thickness. For the model used in this study, the allowed maximum internal pressure is Pmax = 2σyt/Di = 35.84 MPa. In this section, the applied internal pressure ranges from 5 to 35 MPa, and the simulation results are shown in Fig. 8 and 9. For an intact pipe bend, the maximum fluid-induced stress occurs in the intrados of the bend, while the stress level in the extrados is 2–3 times smaller. Note that the predicted erosion rates obtained via the userdefined erosion equation (without stress) and DNV were highly consistent in the elbow section, while the results predicted using DNV diverged significantly from the user-defined equation downstream of the pipe bend. The predicted results in the circumferential direction were

4.2. Effect of fluid velocity Particle velocity is understood to be one of the key parameters governing material erosion [16,21,43]. To investigate the effect of fluid velocity on erosion, different inlet velocities were applied, and the distribution of the erosion rate on the inner surface of the pipe wall was carefully examined. As shown in Fig. 12, the erosion rate increases with increasing fluid velocity - the higher the flow rate, the higher the

Fig. 19. Effect of the particle density on erosion (Vin = 4 m/s, Pout = 15 MPa, ρp = 2650 kg/m3). (a) ρp = 1000 kg/m3 (b) ρp = 1500 kg/m3 (c) ρp = 2000 kg/m3 (d) ρp = 2650 kg/m.3.

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(FD) and pressure gradient (FP), both of which are closely related to the fluid flow. Fig. 13 shows the relationship between the maximum erosion rate and the fluid velocity. For all erosion models investigated, the maximum erosion rate increases exponentially with the increase in fluid velocity. Even when the mechanical effect was considered, the fluid velocity remained the most significant influencing factor. 4.3. Effect of particle size and density The significance of the properties of particles, especially particle size, density, and shape, to their impact on the erosion of material is well established [26]. For the present numerical simulation of erosion, the effect of the particle shape on erosion is reflected by the shape factor Fs (Fs = 1 for angular particles, Fs = 0.53 for semi-rounded particles, and Fs = 0.2 for fully rounded sand particles); in this case, the predicted erosion rate of the pipe wall only differs by a constant. However, real particles have different shapes and sizes, and they will impact the target material and rebound in a highly stochastic manner. Hence, the sharpness of particles plays an important role both in erosion magnitude and rebound. The rebound model mainly affects the rebound behavior of particles and hence govern the secondary impact behavior. Moreover, the irregular shape has a significant impact on the hydrodynamic force acting on the particle, which affects the particle motion. Considering that the focus of this work is the effect of stress on erosion and no substantial experimental data is provided to validate this, the effect of particle shape was not considered in this paper. However, more accurate and robust CFD modeling should be established in the future, in which particle rebound characteristics and sharpness can be considered. To investigate the effect of particle size on erosion, particle diameter (dp) values of 150, 250, 350, and 450 μm were applied. From the results in Fig. 14, it is clear that the erosion rate increases as the particle size increases; the larger the particle diameter, the higher the particle kinematics. These phenomena are more significant when dp < 350 μm. This dependency may be explained based on the dimensionless Stokes number (St), which is defined as [44,57]:

St =

pU

L

,

p

=

2 p dp

18µf

(6)

where τp is the relaxation time of the particle, U is the fluid velocity, L is the flow characteristic length, and μf is the dynamic viscosity of the fluid. In such a flow regime with a high Stokes number, the particle inertia dominates, and the fluid flow effects on particle trajectories are negligible, whereas in slurry flows with low Stokes numbers, sand particles follow the fluid streamlines [44]. The Stokes number increases with the increase in particle diameter (dp) or particle density (ρp), according to Eq. (6). For liquid flow, the dynamic viscosity is much higher than it is for gas flow. In this case, most of the particles follow the streamline and impact the inner surface of the pipe bend in a relatively downstream position. This explains why the predicted hotspot of the erosion scar differs from the test results obtained in Solnordal's [58] research, in which the fluid being conveyed was gaseous. Fig. 14 shows the effect of particle size on the shape of the erosion scar. The area of the erosion effect zone increases as the particle size increases (Fig. 16), and the higher Stokes number causes the early occurrence of erosion in the inlet area of the pipe bend, as the entrainment effect is weaker for larger particles. Fig. 15 shows the maximum erosion rate as a function of particle size, indicating that the maximum erosion rate increases almost linearly with the increase in particle diameter. Such a dependency is

Fig. 20. Effect of surface scar evolution on the erosion rate, Pout = 5 MPa, Vin = 4 m/s.

velocity achieved by particles in the flow. Moreover, it can be seen that the fluid velocity has some impact on the shape of the erosion scar (the shape of the predicted curves through the elbow are different for different inlet velocities), owing to the fact that the hydrodynamic force acting on the particle varies depending on the relative Reynolds number

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Fig. 21. Numerical simulation results with different erosion scars, Pout = 5 MPa, Vin = 4 m/s.

consistent with the study conducted by Desale [59], which indicated that the relationship between the erosion rate and particle size follows the form:

ER

(Particlesize )n

4.4. Process of erosion scar evolution The authors’ previous work proved the significant impact of the surface scar evolution on the predicted erosion rate in a jet impingement test [21]. To investigate the effect of the surface scar evolution on pipe bend erosion, a pipe-wall thickness of 4 mm was chosen to allow for a significant erosion depth and facilitate a more intuitive demonstration. The inlet velocity was 4 m/s, and the outlet pressures were 5 MPa and 15 MPa. Drawing on the modeling technique described in Section 3, the distribution of the erosion rate at different erosion depths (dERmax = 0, 1, 2, and 3 mm, with dERmax denoting the maximum erosion depth) was obtained. Fig. 20 shows the von Mises stress distribution, and erosion rate distribution of the pipe bend in different states of erosion. For the intact pipe bend, the maximum stress distribution occurs in the intrados, and the area of the erosion effect zone is at its maximum. As erosion proceeds, the extrados of the pipe bend is eroded, and the stress hotspot site occurs at the extrados of the pipe bend owing to the concentration of stress at that point. Moreover, the area affected by erosion becomes more localized, owing to the coupling effect of the fluid flow and stress concentration. The results also indicate that the erosion hotspot remains essentially the same as the erosion proceeds. Fig. 21 indicates

(7)

The values of n range from 0.3 to 2.0 in accordance with the differences in material properties, experimental conditions, particle velocity, and particle size. In most cases, n is considered to be 1 [59], although a higher value of n may be used (Fig. 15) when the contribution of stress is taken into account. Fig. 17 shows the relationship between the erosion rate and particle density; in all cases, the fluid density is chosen to be 1000 kg/m3. It can be seen that the area of the erosion effect increases as the particle density rises; this can also be explained by the Stokes number, as the higher the Stokes number, the weaker the entrainment effect. Moreover, the higher the particle density, the higher the kinetic energy is and the greater the impact force, resulting in a greater erosion rate. A nearly linear relationship between the maximum erosion rate and the particle density can be observed (Fig. 18), a dependency that only becomes more significant when the stress acceleration effect is taken into account. The shape of the corresponding erosion scar caused by different particle densities is shown in Fig. 19.

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maximum stress exceeds 0.5σy, as shown in Fig. 22 (c) and (d) and Fig. 23(c) and (d). These results are consistent with the erosion model and test data given by Sun [25] and Wang [21]. The simulation results indicate that the aging of a highly stressed tubular structure is accelerated as erosion proceeds, which can be attributed to the stress acceleration effect arising from stress concentration. Interestingly, the predicted maximum erosion rate obtained via erosion models without consideration of tensile stress initially decreases as erosion proceeds and then increases again (see Figs. 20, 22, and 24), and the change in fluid flow owing to the erosion scar should be responsible. This dependency is more significant under a relatively low operational pressure (5 MPa). However, when the fluid-induced tensile stress on erosion is considered, only a monotonically increasing dependency between the maximum erosion rate and the normalized maximum von Mises stress is observed, as shown in Fig. 24. In this study, the fluid field does not change significantly owing to the limited size of the erosion scar, resulting in a relatively small coupling effect between particle motion and the stress acceleration effect. For a real-world in-service pipe, the wall thickness ranges from 0.6 to 0.9 in, and as such, the depth of an erosion scar may have a greater impact on the fluid flow, producing a much more significant coupling effect. Another interesting phenomenon is that the obtained maximum erosion rate under the same maximum von Mises stress is slightly different for different operational pressures (see Fig. 24, for maximum σMises/σy = 0.3). These differences may arise from the numerical method used to calculate the erosion rate. A different pressure gradient may also contribute to this difference, which is closely related to the pressure gradient force, and change the particle motions. Note that the model mentioned above is suitable for erosion predictions of tubular structures subject to tensile stress and with a low sand concentration. For multiphase flow with a high sand concentration, both particle–fluid interactions and particle–particle interactions should be considered. This will be further investigated in our future work. 5. Conclusion This study investigated the effect of tensile stress, induced by high internal operational pressure, on the erosion of a pipe bend. Moreover, a new method was used to evaluate the evolution of the surface scar. The following conclusions can be drawn from the research findings: (1) A numerical simulation method that was capable of assessing the practical erosion evolution in a typical tubular structure (such as a straight pipe, pipe bend, or plug tee) under high internal pressure, was proposed. The proposed method allows for the consideration of the stress acceleration effect. (2) The predicted erosion rate of a pipe bend with a semi-round particle shape factor (Fs = 0.53) obtained via the proposed erosion equation agrees well with that obtained via the DNV erosion equation, especially in the pipe bend region, and the proposed erosion equation is suitable for the erosion prediction of API X65 pipeline steel. (3) The predicted pipe bend hotspot site is almost identical to that determined by using an alternative erosion equation. Moreover, minor variations were observed during the erosion process. (4) The time-variant profile has some influence on the erosion process, resulting in a more localized erosion damage. This phenomenon is attributed to the coupling effect of the particle movement (affected by fluid flow) and spatial varying stress distribution.

Fig. 22. Effect of surface scar evolution on the erosion rate, Pout = 15 MPa, Vin = 4 m/s (left) stress distribution on the surface of scar, (right) distribution of the erosion rate on the surface of the inner pipe wall.

that the stress on the wall does contribute to the erosion of the pipe bend; however, this contribution is marginal in the downstream section of the straight pipe. The simulation results using a higher internal pressure (15 MPa) are also shown (Fig. 22 and 23); here, the stress acceleration effect is much more significant, especially when the

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Fig. 23. Numerical simulation results with different erosion scars, Pout = 15 MPa, Vin = 4 m/s.

Fig. 24. Maximum erosion rate as a function of the maximum stress.

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Acknowledgement [28]

This project was supported by the National Natural Science Foundation of China (Grant No. 51879189), National Key R&D Program of China (2018YFC0310502), and Guangxi Science and Technology Major Project (Grant No. Guike AA17292007).

[29] [30]

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