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Numerical simulation of solid particle erosion in pipe bends for liquid–solid flow
Numerical prediction of solid particle erosion in pipe bends with liquid–solid flow Wenshan Peng, Xuewen Cao PII: DOI: Reference:
S0032-5910(16)30071-7 doi: 10.1016/j.powtec.2016.02.030 PTEC 11509
To appear in:
Powder Technology
Received date: Revised date: Accepted date:
11 September 2015 12 February 2016 15 February 2016
Please cite this article as: Wenshan Peng, Xuewen Cao, Numerical prediction of solid particle erosion in pipe bends with liquid–solid flow, Powder Technology (2016), doi: 10.1016/j.powtec.2016.02.030
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ACCEPTED MANUSCRIPT Numerical prediction of solid particle erosion in pipe bends with
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Wenshan Peng, Xuewen Cao*
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liquid-solid flow
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College of Pipeline and Civil Engineering, China University of Petroleum, No.66, Changjiang West Road, Qingdao 266580, China
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*Correspondence: Prof. X. Cao ([email protected]), College of Pipeline and Civil Engineering,
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China University of Petroleum, No.66, Changjiang West Road, Qingdao, 266580, China
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Abstract
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Erosion caused by solid particles in pipe bends is one of the major concerns in the oil and gas industry which may result in equipment malfunctioning and even failure. In this work, a two-way
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coupled Eulerian-Lagrangian approach is employed to solve the liquid-solid flow in the pipe bend. Five different erosion models and two particle-wall rebound models are combined to predict the
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erosion rate. The most accurate model is chosen to calculate the effects of a range of parameters on erosion after comparing the predicted results with the experimental data. Further, the relationship between the Stokes number and the maximum erosion location is also assessed. It is found that although all these erosion models generate qualitatively similar erosion patterns, the Erosion/Corrosion Research Center (E/CRC) erosion model with the Grant and Tabakoff particle-wall rebound model produces results that are closest to the experimental data. The sequences of the most influencing factors are obtained: pipe diameter, inlet velocity, bending angle, particle mass flow, particle diameter, and Mean Curvature Radius/Pipe Diameter (R/D) ratio and bend orientation. Additionally, the relationship between Stokes number and the dynamic 1
ACCEPTED MANUSCRIPT movement of the maximum erosion location is presented which can be used to predict the maximum erosion location for different operating conditions. Three collision mechanisms are
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proposed to explain how the changes of Stokes numbers influence the erosion location.
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Keywords: Erosion prediction; Elbow; Two-way coupling; Liquid-solid; Stokes number
1. Introduction
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Erosion caused by solid particles can be considered as a severe problem in the oil and gas industry. Sand is always entrained in the transporting fluid produced from the well. The small
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solid particles flow with the carried fluid and impact the inner wall of the piping, valves and some
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other components. The components face a high risk of solid particle erosion due to the constant
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collision, which may result in equipment malfunctioning and even failure [1-4]. As a complex process, erosion is affected by lots of factors. Particularly in the pipelines, small or subtle change
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of operating conditions can influence the damage due to erosion significantly. Obtaining accurate erosion regularities of different influencing factors is essential for predicting the service life of
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pipelines. Additionally, the accurate erosion prediction can also help in finding the spots where severe erosion is more likely to occur [2]. Many erosion models of solid particles are proposed to calculate the erosion rate of different components. Meng and Ludema [5] carried out a detailed investigation of the erosion models developed previously, and found 28 erosion models were related to solid particle impingement, as well as 33 key parameters that affect erosion rate. Several erosion models mentioned in their work are popularly used in predicting erosion rate for pipe bends, such as the models proposed by Finnie [6], Bitter [7, 8], Neilson and Gilchrist [9], Grant and Tabakoff [10], and Hutchings [11]. The review indicated that each erosion model was developed based on a specific erosion 2
ACCEPTED MANUSCRIPT mechanism and no single predictive equation could be used for practical erosion prediction. Since the 1990s, computational fluid dynamics (CFD) has been widely used for predicting erosion
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caused by solid particles. The CFD method greatly promotes the development of the erosion
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models and several widely used CFD-based erosion models are proposed [12, 13]. Chen et al. [14]
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applied a CFD-based erosion prediction model that was developed by Ahlert [15] and McLaury [16] to predict the relative erosion severity between the elbow and the plugged tee with water/sand
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flow. The particle trajectories and the erosion pattern were analyzed by employing the Grant and Tabakoff [17] particle-wall rebound model. Wood and co-wokers [18] studied the particle
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distributions and particle impingement conditions in particle-laden liquids in Horizontal-vertical
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(H-V) upward pipe bends. The Hashish [19] erosion model was implemented into the CFD
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software to study the effect of impact angles and impact velocities on the pipe. They found that erosion always occurred in specific areas. In another study [20], they performed their erosion
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research on small curvature bend and an upstream straight pipe section. The fluid phase was modeled using slurry flow. The in-plane wall erosion rate calculated by the Hashish erosion model
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[19] was in good agreement with their experimental data. Huang et al. [21] proposed a phenomenological erosion model based on their previous study [22] to calculate the erosion rate of material in slurry flow. The paper suggested that the erosion rate showed a strong dependence on the slurry mean velocity and a weak dependence on pipe diameter and fluid viscosity. The erosion rate has a power-law relation with particle diameter, slurry mean velocity, pipe diameter and liquid viscosity. To investigate the effect of different factors on erosion of liquid-solid flow, various experimental methods have been adopted, such as the slurry pot test [23], jet impingement test [24], Coriolis erosion test [25] and pipe loop test [20,26-29]. The test methods except for the pipe 3
ACCEPTED MANUSCRIPT loop test can only study the erosion of pipe materials and cannot be employed to evaluate the erosion of the real pipelines. However, the loop erosion tests of elbows with liquid-solid flow are
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much fewer than the other tests due to their huge consumption of time and great complexity of
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monitoring. Blanchard et al. [27] studied the effect of Mean Curvature Radius/Pipe Diameter (R/D)
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ratio and particle size on the pipe erosion by using the circulating loop system. They found that the value of the maximum erosion angle was almost the same for elbows under the conditions of
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different size particles or elbow properties. Wood et al. [20] used an experimental loop to explore the erosion rate of the straight and curved ducts. By comparing the results of three different test
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methods, they found that there was a remarkable increase for the erosion of the outermost side
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wall of the bends compared to the innermost side wall and significant erosion occurred at the base
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of the bend. Bourgoyne [28] measured the erosion rate of pipe bend with liquid-solid flow. He studied the effect of R/D ratio, particle mass flow rate and particle velocity on the erosion rate and
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obtained the maximum erosion angle. Although his experimental data of liquid-solid erosion is few, it was referenced by many investigators as a data base to develop erosion equations. Zeng et
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al. [29] investigated the erosion-corrosion (E-C) behavior of an X65 pipeline elbow by using the array electrodes technique. He studied the percentages of the pure erosion rate and found that most of the erosion occurred at the outermost side of the elbow. Most of the currently available CFD-based erosion models and the experimental data of loop tests focus on the pipe bend with gas-solid flow. Studies on the erosion of the pipe bend with liquid-solid flow are relatively few and the accuracy of the erosion model in predicting the erosion of pipe with liquid-solid flow needs further validation. Since abrupt diversion will occur in the elbow section of the pipe which will lead to considerable difference in erosion, this work will look at particulate erosion of a 90° elbow in more detail. The Eulerian-Lagrangian approach is used to 4
ACCEPTED MANUSCRIPT solve the liquid-solid flow. The particle trajectory and the fluctuations of the liquid phase are simulated by the particle-eddy interaction method and the Discrete Radom Walk (DRW) model.
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Five erosion models and two particle-wall rebound models are combined to calculate the erosion
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rate. The particle-fluid interaction is accounted for by using two-way coupling. The erosion
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regularities of the pipe bends and elbows in different flow conditions are analyzed by using the most accurate erosion model. Furthermore, the relationship of Stokes number and the locations
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which are prone to erosion is also studied, and this relationship is explained by three different collision mechanisms.
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2. Numerical modelling
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The Eulerian-Lagrangian method is used in this study. The liquid is treated as a continuous
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phase and solved by the Navier-Stokes equations, while particles are treated as a discrete phase and solved by Newton’s second law. Erosion modeling based on the CFD consists of three steps:
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the continuous phase flow field simulation, particle tracking, and erosion calculation. The first two steps will be described in this section and the last step will be described in the next section.
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2.1. Liquid phase model
The Navier-Stokes equations are employed in this section. The continuity equations and momentum equations are written as:
( v ) 0 t
( v ) ( vv ) p ( ) g S M t
(1) (2)
where ρ is the liquid density, v is the instantaneous velocity vector, p is the pressure, is the stress tensor, g is the body force, S M is the added momentum due to the solid phase. The stress tensor is given by:
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ACCEPTED MANUSCRIPT 2 3
[(v v T ) vI ]
(3)
where μ is the molecular viscosity, I is the unit tensor.
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The standard k-ε model is used to resolve the flow turbulence, and the equations are given as: (4)
( ) ( ui ) 2 [( t ) ] C1 Gk C2 S t xi x j x j k k
(5)
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( k ) ( kui ) k [( t ) ] Gk Sk t xi x j k x j
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where Gk is the generation of turbulence kinetic energy due to the mean velocity gradients, C1ε, C2ε are constants, xi and xj are the spatial coordinates, σk and σε are the turbulent Prandtl numbers
k2
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for k and ε, Sk and Sε are source terms, t C
, σk=1.0, σε=1.3, C1ε=1.44, C2ε=1.92, Cμ =0.09.
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2.2. Disperse phase model
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The particle trajectory is acquired by integrating the motion equation of the particles under the
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Lagrangian coordinates. While setting up the particle tracking and calculating the erosion rate, the following assumptions are made: (1) the injected particles are generally independent of each other
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and the reaction between particles is neglected, (2) the particle breakage is neglected, (3) the modifications of the elbow caused by the particles impaction are neglected. The governing equation of particle motion is proposed according to Newton’s second law: mp
dup FD FP FVM FB dt
(9)
where FD , FP , FVM ,and FB represent the drag force, the pressure gradient force, added mass force and buoyancy force, the expressions of which are given as: FD CD f
d p2 8
| u u p | (u u p )
1 FP d p3P 4 6
(10) (11)
ACCEPTED MANUSCRIPT 1 dup d p3 p 12 dt
(12)
1 FB d p3 ( p ) g 6
(13)
d p | u p u | , mp is the particle mass, up is the particle velocity vector, dp is the
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number, Re s
a a2 3 2 , Res is the particle Reynolds Re s Re s
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where CD is the drag coefficient， CD a1
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FVM
particle diameter, ρp is the density of particles, ρ is the density of fluid, μ is the viscosity of the
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fluid, a1, a2, a3 are constants for the smooth spherical particles given by Morsi and Alexander [30].
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2.3. Coupling between the two phases
The fluid carrier influences the dispersed phase via drag and turbulence, and the particles in
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turn influence the carrier fluid via the reduction in mean momentum and turbulence. The two-way
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coupling is used to solve the interaction between the particles and the liquid. The particle source
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terms are applied to Eqs. (2), (4) and (5) to take these effects into account. 2.3.1. Momentum coupling
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The momentum exchange is computed by examining the change of the particle momentum when it passes through each control volume. This momentum change is computed as:
SM (FD FP FVM FB ) M p t
(14)
where Mp is the mass flow rate of the particles,Δt is the time step. 2.3.2. Turbulence coupling The fluid velocity includes two elements in turbulent flow, the mean velocity and the random fluctuation velocity. The second term can influence particle trajectories. Various methods [31-33] have been proposed to calculate this term. In this paper, the particle-eddy interaction model proposed by Gosman and Ioannides [31] is used to calculate the effect of turbulent fluctuations on 7
ACCEPTED MANUSCRIPT particles. The effect is considered using the DRW model. The turbulent fluctuating velocity that follows a Gaussian distribution is given by:
u ' (u ' ) 2
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where ζ is a random number that obeys normal distribution.
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(15)
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Supposing the local turbulence is isotropic, the local root mean square value of the velocity fluctuation can be calculated by the local turbulent kinetic energy of the flow field: (16)
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( u ' ) 2 2k / 3
Two-way turbulence coupling enables the effect of the change in turbulent quantities due to
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particle damping and turbulence eddies. To take this effect into consideration, the particle source
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terms are included in the k-ε turbulence model, which are in Eqs. (4) and (5). The turbulence
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kinetic energy of liquid phase is modified by the formulation described in [34, 35]. 2.4. Particle impact and rebound behavior
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The particle-wall rebound model is essential in calculating the rebound angle and rebound velocity of particles after impacting the wall. It is also very important for simulating the particle
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trajectories when using the Lagrangian approach. Some commonly used models were developed by Grant and Tabakoff [17], Forder et al. [36], and Sommerfeld and Huber [37]. When a particle impacts the wall, it will lose energy. The rebound velocity is lower than the incident velocity. The restitution coefficient is incorporated to take this effect into consideration. en and et are the restitution coefficients for normal and tangential velocity components which show the change in particle velocity after collision. en and et are written as:
en
up2
et
v p2
u p1
v p1 8
(17)
(18)
ACCEPTED MANUSCRIPT where up1 and up2 are the normal components of the particle velocity before and after impact, vp1 and vp2 are the tangential component of the particle velocity before and after impact, as shown
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in Fig.1.
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In this work, the stochastic particle-wall rebound model by Grant and Tabakoff [17] and the
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non-stochastic particle-wall rebound model by Forder et al. [36] are used with the erosion prediction models to track particles and calculate the erosion. The model proposed by Forder et al.
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[36] is given by:
(19)
et 1 0.78 0.84 2 0.21 3 0.028 4 0.022 5
(20)
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en 0.988 0.78 0.19 2 0.024 3 0.027 4
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where α is the particle incidence angle.
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The model proposed by Grant and Tabakoff [17] is given by: (21)
et 0.988 1.66 2.11 2 0.67 3
(22)
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3. Erosion models
en 0.993 1.76 1.56 2 0.49 3
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Alternative methods of predicting erosion rates have been suggested by many investigators. The solid particle erosion models are in the general form:
ER Kf ( )u np
(23)
where ER is the erosion rate of the target. K is a constant depending on the target property, particle shape, particle hardness and some other factors, f(α) is a dimensionless function of the impact angle, n is a material dependent index. The erosion rate is defined as the wall mass loss per unit area and per unit time (kg/m2s).The erosion is finally given as the penetration rate (m/s), which is evaluated by dividing the erosion rate (kg/m2s) by the density of pipe wall material (kg/m3). As the penetration rate calculated in this study is very small, we use a smaller unit (nm/s) 9
ACCEPTED MANUSCRIPT to represent the erosion throughout this work. Five frequently used erosion models are analyzed and compared in this section.
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3.1. Ahlert erosion model
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Ahlert [15] developed an erosion model for AISI 1018 steel. This model was based on a series
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of direct impact experiments for different particle shapes and impact angles. The erosion model is given by:
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ER A( BH )0.59 Fsu np f ( )
f ( ) a 2 b
0
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f ( ) x cos2 sin( w ) y sin 2 z 0
(24) (25) (26)
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where A is a constant, A=15.59×10-7 for carbon steel, α is the particle impact angle, α0 is the
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transition angle, normally set as15°, n=1.73, a, b, w, x, y, z are listed in Table 1. Fs is the particle shape coefficient (Fs=1.0 for sharp particles, 0.53 for semi-rounded particles and 0.2 for fully
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rounded particles). BH is the Brinell’s Hardness of the target material. 3.2. DNV erosion model
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DNV [38] proposed an erosion model for various target materials based on a large number of experimental data. This model can predict the erosion for steel, epoxy, titanium, vinyl ester. As the model is relatively simple, it can be used in the CFD software easily.
ER Cf ( )u np
(27)
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f ( ) ( 1)i 1 Ai i
(28)
i 1
where C is a constant determined by the target material. For steel pipes, C=2.0×10-9, n is the velocity exponent, for steel pipes, n=2.6. Values of Ai are given in Table 1.
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ACCEPTED MANUSCRIPT 3.3. E/CRC erosion model The Erosion/Corrosion Research Center (E/CRC) at the University of Tulsa [13] developed the
given according to Zhang et al. [4] as:
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f ( ) Ri i i 1
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ER C ( BH )0.59 FS u np f ( )
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E/CRC models to predict the erosion of elbows, tees and some other pipe fittings. The model is
(29) (30)
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where C is the constant and equal to 2.17×10-7 for carbon steel, n=2.41. Values of Ri are listed
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in Table 1.This model is similar to the Ahlert erosion model, but the constant C is much smaller than A. The E/CRC erosion model is a little more complicated than the DNV erosion model
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because the particle hardness and the particle sharpness are added into consideration.
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3.4. Neilson and Gilchrist erosion model
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Neilson and Gilchrist [9] proposed an erosion model based on the impact angle in the basis of Finnie and Bitter’s study. They developed two equations for particles erosion of large impact
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angles and small impact angles and suggested that the total erosion rate was equal to the summation of erosion due to deformation and cutting mechanisms. This model was derived based on the experimental results:
ER
ER
u 2p cos2 sin 2 C u 2p cos2 2 C
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u 2p sin 2 2 D
u 2p sin 2 2 D
0
(31)
0
(32)
where α0 is the transition angle, normally set as 45°, εC is the cutting coefficient, set as 3.332× 107, εD is the deformation coefficient, set as 7.742×107.
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ACCEPTED MANUSCRIPT 3.5. Oka et al. erosion model Oka et al. [39, 40] developed an erosion model by taking into account many parameters that
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affect the material erosion. An angle function combined with two terms was proposed to calculate
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the erosion damage. This model considers more influencing factors than the above models, such as
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the particle diameter, the reference diameter and the impact speed of the reference particle, thus it has a more broad application. This erosion model is expressed as:
up
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ER 1.0 109 wk0 f ( )( Hv )k1 (
V
'
) k2 (
dp d
) k3
(33) (34)
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f ( ) (sin )n1 [1 Hv(1 sin )]n2
'
where ρw is the density of target material, Hv is the Vickers hardness of the target material, dp
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is the particle diameter, d’ is the reference diameter, V’ is the impact speed of the reference particle,
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k0, k1, k2, k3, n1, n2, d’, V’ are listed in Table 1.
4.1. Case description
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4. CFD modeling
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The commercial software ANSYS FLUENT is the CFD solver used for the numerical simulations. A database of the experiment conducted by Zeng et al. [29] is employed in this work to investigate the performed erosion models. Zeng et al. studied the erosion of an elbow using a circulating loop system. 38 sets of experimental data were gathered on the same elbow. The test piece was a 90°elbow with a diameter of 50 mm and a curvature radius of 76.9 mm, which can be seen in Fig.2. In order to better represent the flow at the specimen location, a 1m (20D) vertical pipe upstream and a 0.5m (10D) horizontal pipe downstream of the elbow were used. The flow conditions are shown in Table 2. 4.2 Computational mesh
Fig.3 shows the 3-D computational domain used in this work. Generally, meshing consists of 12
ACCEPTED MANUSCRIPT two parts: surface meshing and volume meshing. The surface grids are carefully generated due to its important effect on the quality of resulting volume grids. In order to obtain an accurate flow
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field near the pipe wall, a finer grid scheme in the boundary layer must be used. The boundary
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layer near the pipe inner wall has five-layer grids (the first percent of the cell height of the first
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layer is 20% and the growth factor is 1.2). The structured grid is used to mesh the surface of the cross-section. Finally, the hexahedral structured mesh is adopted to mesh the whole volume. In
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present study, grid independency study has shown that larger number of grids will not necessarily influence the accuracy of the cases when the grid quantity achieves a certain number. The grid
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number used in the base case is approximately 850,000. 4.3. Boundary conditions
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In this work, the solid particles are injected uniformly at the pipe inlet at the same speed as
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the water. The working pressure is set as 1.01×105 Pa and the temperature is set as 25 ℃. The
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turbulence intensity is set at 3.5%. Additionally, wall roughness height is set as 10μm which is consistent with the study of Zeng et al. [29]. The roughness constant is set as the default value of
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0.5. The wall boundaries are set as ‘reflect’ and the outlet boundary is set as ‘escape’. 4.4. The simulation procedure
The SIMPLEC algorithm is used to solve the pressure and velocity field in order to improve convergence. The standard discretization schemes are used for the pressure terms and the second order upwind discretization schemes are used for the convection terms and divergence terms. The convergent criteria for all the steady simulations are set as that the residual in the control volume for each equation is smaller than 10-5 or the number of iterations reaches to 5000. The number of continuous phase iterations per discrete phase model (DPM) iteration is set as 5. A total of 10260 particles will be tracked in the simulation. The simulations are executed in the Intel (R) Xeon (R) 13
ACCEPTED MANUSCRIPT with specifications of CPU E5-2690 2.90GHz and 16.0 GB RAM with windows server 2008 platform. About 6 CPU hours are consumed for one case.
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5. Results and discussion
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5.1. Flow model verification
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Since the flow field simulation is the first step of the erosion calculation, it is very important to the accuracy of the erosion predictions. In order to verify the flow modeling capabilities, a
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verification study is performed on a 90°elbow, as shown in Fig.4. By comparing the predicted velocity profiles with the experimental data of Enayet et al. [41], the accuracy of the flow solution
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can be assessed. Enayet et al. studied the flow field of a 90° pipe bend with the R/D ratio equal to 2.8 using the Laser Doppler Velocimetry (LDV) technique. They obtained the velocity profiles of
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different sections of an elbow for a turbulent flow case with Reynolds number equal to 43000. Measurements were made in planes perpendicular to the flow direction at 30°,60° and 75°。
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Comparison of predicted velocity profiles and experimental velocity profiles is shown in Fig.4. There is good agreement between the predictions and the experimental data. It is worth noticing
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that the k-ε model can successfully predict the changes of flow in the pipe bend. The velocity changes more dramatically near the inside wall than that near the outside wall. This may be attributed to the effect of the secondary flow at the elbow. These secondary flows can significantly influence the erosion profiles. The relationship between erosion profiles and the secondary flow are described in the following sections. 5.2. Comparison among erosion models Fig.5 shows the penetration rate around the elbow obtained by five CFD-based erosion models and the experimental data by Zeng et al. [29]. It can be noticed that the Ahlert erosion model results in much higher penetration rate compared to the experimental data. The penetration rate 14
ACCEPTED MANUSCRIPT calculated by Oka et al. erosion model and Neilson and Gilchrist erosion model also over-predict the penetration rate. The reason may be that the flow conditions and the particle property used in
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Zeng’s experiment are not similar to the conditions in the numerical models. Specificity of the
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material used in Neilson and Gilchrist experiment may be also a contributory factor to the
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over-prediction. Fig.5 shows that both the predictions obtained by the DNV erosion model and by the E/CRC erosion model show good agreement with the experimental data. As the erosion data
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from Zeng et al. [29] was only about the erosion around the elbow under a certain experimental condition, more experimental data is needed to further verify which model can produce results that
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are closest to the experimental data. Bourgoyne [28] studied solid particle erosion in diverter
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systems and several experimental data were gathered for sand erosion in liquid flow. Although
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only several data are available, they can help us find the most accurate erosion model after comparing the numerical erosion rate to the experimental data. The flow conditions of the
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experiments conducted by Bourgoyne[28] are described in Table 3. As shown in Fig.6, the ratios of predicted penetration to experimental penetration lie in the
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range of 1.01-1.82 when the E/CRC model and the Grant and Tabakoff model are applied, which are smaller than the other three sets of predictions. It indicates that the prediction calculated by E/CRC erosion model with the Grant and Tabakoff particle-wall rebound model yields the best agreement with the experimental data. Thus, in the following sections, the effect of various influencing factors such as inlet velocity, particle diameter, pipe diameter, particle mass flow rate, bend orientation, bending angle and R/D ratio will be assessed by using this model. The CFD simulations are performed under different flow conditions as described in Table 4.
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ACCEPTED MANUSCRIPT 5.3. Effect of inlet velocity on predicted penetration rate Fig.7 shows the effect of inlet velocity on predicted penetration rate. Fig.7a indicates that the
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penetration rate increases exponentially as the inlet velocity increases which means that the
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particle velocity has a great influence on the penetration rate. A larger inlet velocity will result in
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larger impact energy and as a result, the penetration rate is higher. Significant difference of the first erosion peak is observed in Fig.5b, the maximum erosion under low inlet velocity appears
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earlier than that under high inlet velocity. For inlet velocity of 5m/s, the first remarkable erosion peak appears at 67°.However, it rises to 86° for inlet velocity of 25m/s. Fig.7b also displays that
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under the same inlet velocity, the penetration rate increases as the angle of the elbow increases.
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The maximum penetration location appears at 90° of the elbow.
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5.4. Effect of particle diameter on predicted penetration rate Fig.8 shows the effect of particle diameter on predicted penetration rate. Fig.8a suggests that
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as the particle diameter increases, the penetration rate first decreases then increases, with the minimum value occurring at 150μm. Thus, 150μm is the critical diameter for the erosion. The
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redirection of the liquid in the elbows has great influence on the particles, especially for the small particles. Fig.8a presented that the penetration rate in the condition of 50μm is even higher than that of 200μm. The reason for this special phenomenon may be that the dominant force for the particles with different diameter is changed. For small particles, the intense secondary flow in the elbows causes the particles to impact the side wall of the elbow (region p1) and results in severe erosion (region s1), as shown in Fig.9. For large diameter particles, the significance of the secondary flow effect on the particles decreases with respect to the inertia force. The inertia force drives the particles impact directly on the elbow (region p3) and cause severe erosion in the outside wall (region s3). For the medium diameter particles, both the secondary flow effect and the 16
ACCEPTED MANUSCRIPT inertia force are remarkable, these two mechanisms separate the particles into two major paths (region p2) and cause a large scale erosion scar on the inner surface of the elbow (region s2). Since
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the particles are separated into two parts by the two mechanisms, the penetration rate maybe the
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lowest compared to the other two conditions.
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It can also indicate that as the particle diameter increases, the maximum penetration rate will occur at a lower angle, as depicted in Fig.8b. This is because the trajectories of large particles are
then impact the outer wall at a lower angle.
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mainly controlled by inertia force. This force will drive the large particles flow in straight path and
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5.5. Effect of pipe diameter on predicted penetration rate
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Fig.10 shows the effect of pipe diameter on predicted penetration rate. In Fig.10a, the
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penetration rate decreases sharply as the pipe diameter increases. The penetration rate decreases from 0.681nm/s to 0.0122nm/s as the pipe diameter increases from 40mm to 400mm, which
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reduces more than 50 times. When the pipe diameter exceeds 400mm, the penetration rate will decrease slowly and stabilize at a very low level. This effect may be attributed to the fact that the
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pipe with large diameter has a relatively larger impact surface. Since the particle mass flow rate is kept at 0.02kg/s for these cases, the erosion rate can be defined as the penetration rate per unit area (nm/s/m2) to make more sense for this comparison. For example, the value of erosion rate is 5.81×10-9 nm/s/m2 for the elbow with a diameter of 400mm and 2.78×10-9 nm/s/m2 for the elbow with a diameter of 500mm. The discrepancy between them is not remarkable. Moreover, the maximum penetration appears at between 75° and 90° for all pipe diameters in Fig.10b. 5.6. Effect of particles mass flow rate on predicted penetration rate Fig.11 shows the effect of particles mass flow rate on predicted penetration rate. The 17
ACCEPTED MANUSCRIPT penetration rate increases linearly as the particle mass flow increases, as shown in Fig.11a. This can be explained that as the mass flow rate increases, more solid particles will impact the pipe
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wall at a time and result in higher penetration rate. Under the same particle mass flow rate, the
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penetration rate increases slowly with the increasing of the degree, as shown in Fig.11b. Zheng et
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al. [42] examined the effect of particle mass flow rate and found that the penetration rate will no longer increase when the mass flow rate increases to a certain value. As the mass flow rate
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increases, the loss of the momentum from the collisions between particles becomes dramatically. Thus, the effect of the particle mass flow decreases at high particle mass flow rates. This effect is
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not shown in our results, however. This may be attributed to the fact that we use the two-way
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coupling in this work and the particle-particle interactions are neglected. Further under-predictions
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are anticipated if the effect of particle-particle interactions is taken into consideration in the numerical simulations. For high loading flow, the four-way coupling which considers the
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particle-particle interactions will be necessary. 5.7. Effect of bend orientation on predicted penetration rate
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Fig.12 shows the effect of bend orientation on predicted penetration rate. It suggests that there is no significant difference in the penetration rate when the bend orientation changes. The erosion regularities of the elbows with different bend orientation are similar. The maximum erosion locations for these elbows all occur at about 85°. 5.8. Effect of bending angle on predicted penetration rate Fig.13 shows the effect of bending angle on predicted penetration rate. The penetration rate increases as the bending angle increases, as shown in Fig.13a.On one hand, due to geometrical shape of these three elbows, the impact angle of the particles at the elbow wall of the 90°elbow and 180°elbow is larger than that of the 45°elbow. The maximum impact angle for 90°elbow 18
ACCEPTED MANUSCRIPT and 180°elbow is 60°and changes to 45° for 45°elbow. When the impact angle increases from 45° to 60°, the value of f(α) in Eq.(30) first increases from 1.316 to 1.325 then decreases to 1.296. The
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average value of f(α) in the range of 45°to 60°is larger than the value of f(α) at 45°.Thus, the value
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of f(α) in large bending angle elbow is larger than that in small bending angle elbow. On the other
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hand, as the maximum velocity area of the large bending angle pipe bend is larger than that of small bending angle pipe bend in the elbow region, the particles in large bending angle pipe bend
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will get a larger impact velocity after crossing through this region, as shown in Fig.14. Since f(α) and particle impact velocity directly increase the erosion rate of pipe bend which can be seen in
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Eqs.(29) and (30), the large bending angle elbow experiences more severe erosion than small
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bending angle elbow.
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Additionally, the particles in larger angle elbows experience multiple collisions with the inner wall due to the fact that the length of large angle elbow is greater than that of small angle elbow.
shown in Fig13b.
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As a result, the 180° elbow presents more penetration peaks than 90° elbow and 45° elbow, as
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5.9. Effect of R/D ratio on predicted penetration rate Fig.15 shows the effect of R/D ratio on predicted penetration rate. The penetration rate decreases slowly as the R/D range from 1.5 to 4, and then levels off slightly as the R/D ratio is greater than 4, as shown in Fig.15a. Therefore, there is an economical ratio in the engineering practice. Since the diameters of the pipe bends are set as 40mm in these runs, the larger R/D ratio makes the pipe path longer and as a result, the solid particles flow more smoothly and cause less impact on the wall. The secondary flow influences the penetration rate more significantly as the R/D ratio increases. The secondary flow will occur earlier and drive the particles to impact the side wall of the elbow. As a consequence, the erosion at the outermost of the elbow becomes less 19
ACCEPTED MANUSCRIPT obvious as shown in Fig.15b. Thus, it is beneficial to use long-radius elbows in the engineering practice.
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5.10. The relationship between Stokes number and the erosion locations
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The movement of particles through elbows is governed by three forces: particle inertia force,
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drag force exerted by surrounding water and secondary flow. The particle inertia force maintains the particle move in the tangential direction. The drag force motivates particles flow along the
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water streamline, and the secondary flow forcing solid particles move towards the outer wall. Stokes number as the ratio of particle relaxation time and fluid characteristic time can reflect the
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magnitude of the particle inertia force and the drag force and it is a dimensionless number for
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representing the curvilinear movement of the solid particle. Thus, Stokes number is adopted to
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The form of Stokes number is:
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evaluate the relationship between these three mechanisms and their effect on the pipe bend erosion.
St
p d p2u 18 D
(35)
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In order to briefly describe the severity of the erosion in elbows, the ‘erosion degree’ is imported in this section. The erosion degree includes three levels. The high erosion degree represents that the elbow experiences severe erosion and the low erosion degree represents that the elbow experience slight erosion, and the medium erosion degree is between them. It can be seen from Fig.16 that there are two dynamic severe erosion locations which are location A and location B respectively. The maximum erosion location will dynamic change as the Stokes number increases. Location A is exposed to greater erosion when the Stokes number is small. However, as Stokes number increases, this maximum erosion area will move to location B. When Stokes number is small, the drag force is dominant. The secondary flow plays an important role in the 20
ACCEPTED MANUSCRIPT movement of particles. The secondary flow first drives the particles in the circumferential direction from the outer side to the inner side of the elbow and then drives the particles to the side
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wall of the elbow and as a result, causing severe erosion in this area. For a larger Stokes number,
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the inertia force plays a larger role on the particles movement. The particles have enough
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momentum to flow across eddies, thus the fluid velocity and the fluid flow direction have a smaller influence on the particles. As a result, the solid particles deviate from the fluid streamlines
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and impact on the outer wall.
A dimensionless number is defined to analyze the effect of Stokes number on the maximum
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erosion rate of location B. This number describes the relative magnitudes of the maximum erosion rate of this area and the maximum erosion rate of the whole pipe bend, the number is given as:
Max.ERelbow Max.ER pipe
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(36)
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Fig.17 shows the relationship between β and the Stokes number. As the Stokes number increases, the number β first increases and then stabilizes at 1.0. The number β will no longer
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change when Stokes number is larger than 1.472. This indicated that 1.472 is a division value for Location A and Location B. When the Stokes number is smaller than 1.472, the maximum erosion will occur at Location A, otherwise, the maximum erosion will occur at Location B. The sample particle trajectories and velocity vectors for standard pipe bend are shown in Fig.18.There are three paths of particles, namely a-b-c-d, e-f-g and h-i-j. These three paths are controlled by inertia force, drag force and secondary flow, that is, they are influenced by the Stokes number. Location A can be explained by path a-b-c-d, as well as Location B can be explained by path e-f-g and h-i-j. For small Stokes numbers, the particles readily flow along the streamlines, thus these particles will change their flow direction due to the secondary flow which 21
ACCEPTED MANUSCRIPT first appears at location a, as shown in Fig.18. The secondary flow drives these particles to leave from location a and collide at location b. For the fluid having the largest velocity in the inner side
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of the elbow, the velocity of the particles become larger from location a to location b. The particles
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impact the inner side wall (Location A) with great kinetic energy and then cause severe erosion.
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The probability of rebounding to location c or d after colliding at location b depends on the particle impact energy, the flow field of the bend, the wall roughness, and the particle dispersion.
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In Fig.18, the erosion rate at location c and d are much lower than that at location a and location b. This is mainly because the flow field at downstream of the pipe bend is much smoother than that
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at the elbow. As the particle paths are almost parallel to the wall surface, the particle-wall impacts
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are at a much lower impact angle and the impact velocity perpendicular to the surface is also small.
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Both these effects decrease the erosion.
For larger Stokes numbers, the particles will move more independently and impact the
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outermost wall of the elbow directly which performs as path e-f-g and path h-i-j. For the particles with larger momentum, they will impact location e directly and as a result, the larger impact
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energy leads to heavy erosion at this location. Furthermore, after rebounding from location e, the particles will collide at location f and location g. However, the erosion rate at location f and location g is much lower than that at location e due to the smaller particle impact angle and the smaller impact velocity perpendicular to the surface. For particles in smaller velocity, the particles will flow from location h, and experience a sliding collision to location i, finally rebound to location j at a small rebound angle. The direct collisions and the sliding collisions determined these two paths and the erosion degree of location B. 6. Conclusions An Eulerian simulation with the k-ε turbulence model was used to simulate the liquid flow in a 22
ACCEPTED MANUSCRIPT 90° pipe bend, while the Lagrangian particle tracking method was employed for calculating the particle motion. The liquid-solid interaction was also taken into account. Five erosion models with
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two particle rebound models were combined to predict the erosion rate and the maximum erosion
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location. Based on the simulation results, it was concluded that the combination of the E/CRC
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erosion model with Grant and Tabakoff particle-wall rebound model is the most suitable combination for the erosion prediction because it provided the best agreement with the
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experimental data of Bourgoyne[28] and Zeng et al. [29].
The effects of different parameters on penetration rate were studied. The penetration rate
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increases as the inlet water velocity, mass flow rate and bending angle increase. On the other hand,
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the penetration rate decreases as the pipe diameter and R/D ratio increase. The bend orientation
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and flow direction have little effect on the change of the penetration rate. Particularly, it can be concluded that the penetration rate first decreases then increases, with the minimum value
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occurring at the particle diameter of 150μm. The penetration rate of small particles which are smaller than 150μm can even result in higher penetration rate than that of larger particles. Since in
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the CFD runs, only one influencing parameter is different at a time, the effect of that single parameter can be determined. The sequences of the most influencing parameters are: pipe diameter, inlet velocity, bending angle, particle mass flow, particle diameter, and the R/D ratio and bend orientation. Furthermore, the relationship between Stokes number and the dynamic movement of the maximum erosion location is presented. Erosion mainly occurs at the outermost side of the elbow and the side walls of the downstream straight pipe close to the elbow outlet. The most severe erosion location changes as Stokes number varies. For smaller Stokes numbers the maximum erosion location always occurs at the inner side walls of the elbow, while for larger Stokes 23
ACCEPTED MANUSCRIPT numbers, this location always occurs at the outermost side of the elbow. The Stokes number also influences the particle trajectories, thus, three particle paths are investigated to explain the erosion
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location. The results indicates that the erosion of the side wall is mainly caused by the secondary
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flow driven collision, while the erosion of outermost side of the wall is caused by a combined
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effect of direct collision and sliding collision. Acknowledgments
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The authors would like to acknowledge the financial support of National Natural Science Foundation of China (NO.51274232) and the Fundamental Research Funds for the Central
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Universities of China (No.15CX06070A).
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References
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[1] C. B. Solnordal, C. Y. Wong, J. Boulanger, An experimental and numerical analysis of erosion caused by sand pneumatically conveyed through a standard pipe elbow. Wear 336 (2015) 43-57.
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[2] G. C. Pereira, F. J. de Souza, D. A. de Moro Martins, Numerical prediction of the erosion due to particles in elbows. Powder Technol. 261 (2014) 105-117.
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[3] M. Jafari , Z. Mansoori, M. S. Avval, G. Ahmadi, A. Ebadi, Modeling and numerical investigation of erosion rate for turbulent two-phase gas–solid flow in horizontal pipes. Powder Technol. 267(2014)362-370. [4] Y. Zhang, E. P. Reuterfors, B. S. McLaury, S. A. Shirazi, E. F. Rybicki, Comparison of computed and measured particle velocities and erosion in water and air flows. Wear 263 (2007) 330-338. [5] H. C. Meng, K. C. Ludema, Wear models and predictive equations – Their form and content. Wear 181–183 (1995) 443–457. [6] I. Finnie, Erosion of surfaces by solid particles. Wear 3 (1960) 87–103. [7] J. Bitter, A study of erosion phenomena part I. Wear 6 (1963) 5-21. [8] J. Bitter, A study of erosion phenomena: part II. Wear 6 (1963) 169-190. 24
ACCEPTED MANUSCRIPT [9] J. H. Neilson, A. Gilchrist, Erosion by a stream of solid particles. Wear 11 (1968) 111-122. [10] G. Grant, W. Tabakoff. An experimental investigation of the erosive characteristics of 2024 aluminum alloy. No.
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73-37. Cincinnati: Department of Aerospace Engineering, University of Cincinnati, 1973.
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[11] I. M. Hutchings, R. E. Winter, J. E. Field. Solid particle erosion of metals: the removal of surface material by
Sciences. Vol. 348. No. 1654. The Royal Society, 1976.
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spherical projectiles. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering
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[12] D. O. Njobuenwu, M. Fairweather. Modelling of pipe bend erosion by dilute particle suspensions. Comput. Chem. Eng. 42 (2012)235-247.
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[13] M. Parsi, K. Najmi, F. Najafifard, S. Hassani, B. S. Mclaury, S. A. Shirazi, A comprehensive review of solid
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particle erosion modeling for oil and gas wells and pipelines applications. J. Nat. Gas Sci. and Eng. 21
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(2014) 850–873.
[14] X.Chen, B. S.McLaury, S. A. Shirazi, Numerical and experimental investigation of the relative erosion
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severity between plugged tees and elbows in dilute gas/solid two-phase flow. Wear 261 (2006) 715-729. [15] K. R. Ahlert, Effects of particle impingement angle and surface wetting on solid particle erosion of AISI 1018
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steel. Ph.D. Thesis, Department of Mechanical Engineering, The University of Tulsa, 1994. [16] B. S. McLaury, Predicting solid particle erosion resulting from turbulent fluctuations in oilfield geometries, Ph.D. Dissertation, The University of Tulsa, OK, 1996. [17] G. Grant, W. Tabakoff, Erosion prediction in turbomachinery resulting from environmental solid particles. J. Aircraft 12(1975)471-478. [18] R. J. K. Wood, T. F. Jones, N. J. Miles, J. Ganeshalingam, Upstream swirl-induction for reduction of erosion damage from slurries in pipeline bends. Wear 250 (2001) 770-778. [19] M. Hashish, An improved model of erosion by solid particle impact, in: Proceedings of the 7th International Conference on Erosion by Liquid and Solid Particle, Cambridge, 1987, pp. 66/1–66/9. 25
ACCEPTED MANUSCRIPT [20] R. J. K. Wood, T. F. Jones, J. Ganeshalingam, N. J. Miles, Comparison of predicted and experimental erosion estimates in slurry ducts. Wear 256 (2004) 937-947.
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[21] C. Huang, P. Minev, J. Luo, K. Nandakumar, A phenomenological model for erosion of material in a horizontal
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slurry pipeline flow. Wear 269 (2010) 190-196.
of materials in jet flow. Powder Technol. 187 (2008) 273-279.
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[22] C. Huang, S. Chiovelli, P. Minev, J. Luo, K. Nandakumar, A comprehensive phenomenological model for erosion
solid–liquid mixture. Wear 259 (2005) 196-202.
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[23] G. R. Desale, B. K. Gandhi, S. C. Jain, Improvement in the design of a pot tester to simulate erosion wear due to
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[24] E. Mahdi, A. Rauf, E. O. Eltai, Effect of temperature and erosion on pitting corrosion of X100 steel in aqueous
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silica slurries containing bicarbonate and chloride content. Corros. Sci. 83 (2014) 48-58.
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[25] Y. Xie, H. M. Clark, H. M. Hawthorne, Modelling slurry particle dynamics in the Coriolis erosion tester. Wear 225 (1999) 405-416.
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[26] C. A. Shook, M. McKibben, M. Small, Experimental investigation of some hydrodynamic factors affecting slurry pipeline wall erosion. Can. J. Chem. Eng. 68 (1990) 17-23.
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[27] D. J. Blanchard, P. Griffith, E. Rabinowicz, Erosion of a pipe bend by solid particles entrained in water. J. Manuf. Sci. Eng. 106 (1984) 213-217. [28] A. T. Bourgoyne Jr, Experimental study of erosion in diverter systems due to sand production. In SPE/IADC Drilling Conference. Society of Petroleum Engineers,1989. [29] L. Zeng, G. A. Zhang, X. P. Guo, Erosion–corrosion at different locations of X65 carbon steel elbow. Corros. Sci. 85 (2014) 318-330. [30] S. Morsi, A. J. Alexander, An investigation of particle trajectories in two-phase flow systems. J. Fluid. Mech. 55 (1972) 193-208. [31] A. D. Gosman, E. Ioannides, Aspects of computer simulation of liquid-fueled combustors. J. Energ.7 (1983) 26
ACCEPTED MANUSCRIPT 482-490. [32] G. Klose, B. Rembold, R. Koch, S. Wittig, Comparison of state-of-the-art droplet turbulence interaction models
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for jet engine combustor conditions. Int. J. Heat. Fluid. Fl. 22 (2001) 343-349.
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[33] C. Marchioli, A. Soldati, Mechanisms for particle transfer and segregation in a turbulent boundary layer.
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J. Fluid. Mech. 468 (2002) 283-315.
[34] A. A. Amsden, P. J. O'rourke, T. D. Butler, KIVA-II: A computer program for chemically reactive flows with
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sprays (No. LA-11560-MS). Los Alamos National Lab., NM (USA), 1989.
[35] G. M. Faeth, Spray atomization and combustion.Technical Report 1986-136, AIAA,1986.
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control valves. Wear 216 (1998) 184-193.
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[36] A. Forder, M. Thew, D. Harrison, A numerical investigation of solid particle erosion experienced within oilfield
Flow 25 (1999) 1457-1489.
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[37] M. Sommerfeld, N. Huber, Experimental analysis and modelling of particle-wall collisions. Int. J. Multiphase.
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[38] D. N. Veritas, Recommended Practice, Erosion Wear of Piping Systems. DNV RP O, 501, 2007. [39] Y. I. Oka, K. Okamura, T., Yoshida Practical estimation of erosion damage caused by solid particle impact. Part 1:
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Effects of impact parameters on a predictive equation. Wear 259 (2005) 95–101. [40] Y. I. Oka, T. Yoshida, Practical estimation of erosion damage caused by solid particle impact. Part 2: Mechanical properties of materials directly associated with erosion damage. Wear 259 (2005) 102–109. [41] M. M. Enayet, M. M. Gibson, A. M. K. P. Taylor, M. Yianneskis, Laser-Doppler measurements of laminar and turbulent flow in a pipe bend. Int. J. Heat. Fluid. Fl. 3 (1982) 213-219. [42] Y. G. Zheng, H. Yu, S. L. Jiang, Z. M. Yao, Effect of the sea mud on erosion-corrosion behaviors of carbon steel and low alloy steel in 2.4% NaCl solution. Wear 264 (2008) 1051-1058.
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ACCEPTED MANUSCRIPT Table captions Table 1 Values of parameters in solid particle erosion models.
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Table 2 Experimental conditions from Zeng et al. [29].
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Table 3 Experimental conditions from Bourgoyne[28].
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Table 4 Conditions of the CFD runs.
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Figure captions
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Fig.1. Definition of the impact procedure of a particle: (a) straight wall; (b) curved wall.
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Fig.2. Computational geometry used in the simulation.
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Fig. 3. Computational mesh used in the simulation.
Fig.4. Comparison of predicted velocity profiles with experimental data of Enayet et al.[41].
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Fig.5. Comparison of predicted and experimental erosion rate along elbow curvature angle for the five erosion models with different particle-wall rebound models.
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Fig.6. Comparison of measured erosion from Bourgoyne [28] with different erosion models.
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Fig.7. Effect of inlet velocity on predicted penetration rate (CFD runs 1-5): (a) penetration rate of the whole pipe
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bends; (b) penetration rate of the elbows.
Fig.8. Effect of particle diameter on predicted penetration rate (CFD runs 2, 6-12): (a) penetration rate of the
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whole pipe bends; (b) penetration rate of the elbows. Fig.9. Erosion profiles and particle trajectories of pipe bend with different particle diameter:(a) 50μm；
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(b)150μm;(c)300μm.
Fig.10. Effect of pipe diameter on predicted penetration rate (CFD runs 2, 13-17): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. Fig.11. Effect of particles mass flow rate on predicted penetration rate (CFD runs 2, 18-22): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. Fig.12. Effect of bend orientation on predicted penetration rate (CFD runs 2, 23-25): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. Fig.13. Effect of bending angle on predicted penetration rate (CFD runs 2, 26 and 27): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. 29
ACCEPTED MANUSCRIPT Fig.14. Flow field of the pipe bends in different bending angle. Fig.15. Effect of R/D ratio on the predicted penetration rate (CFD runs 2, 28-34): (a) penetration rate of the
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whole pipe bends; (b) penetration rate of the elbows.
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Fig.16. The erosion location for the pipe bends as the Stokes number changes. Location A is the side wall of the
Fig.17. The relationship between β and the Stokes number.
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elbow, location B is the outermost side of the elbow close to the elbow outlet.
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Fig.18. Sample particle trajectories and velocity vectors for standard pipe bend.
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ACCEPTED MANUSCRIPT Figure captions
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Fig.1. Definition of the impact procedure of a particle: (a) straight wall; (b) curved wall.
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Fig.2. Computational geometry used in the simulation.
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Fig. 3. Computational mesh used in the simulation.
Fig.4. Comparison of predicted velocity profiles with experimental data of Enayet et al.[41].
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Fig.5. Comparison of predicted and experimental erosion rate along elbow curvature angle for the five erosion models with different particle-wall rebound models.
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Fig.6. Comparison of measured erosion from Bourgoyne [28] with different erosion models. Fig.7. Effect of inlet velocity on predicted penetration rate (CFD runs 1-5): (a) penetration rate of the whole
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pipe bends; (b) penetration rate of the elbows. Fig.8. Effect of particle diameter on predicted penetration rate (CFD runs 2, 6-12): (a) penetration rate of the
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whole pipe bends; (b) penetration rate of the elbows. Fig.9. Erosion profiles and particle trajectories of pipe bend with different particle diameter:(a) 50μm；
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(b)150μm;(c)300μm.
Fig.10. Effect of pipe diameter on predicted penetration rate (CFD runs 2, 13-17): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. Fig.11. Effect of particles mass flow rate on predicted penetration rate (CFD runs 2, 18-22): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. Fig.12. Effect of bend orientation on predicted penetration rate (CFD runs 2, 23-25): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. Fig.13. Effect of bending angle on predicted penetration rate (CFD runs 2, 26 and 27): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. 31
ACCEPTED MANUSCRIPT Fig.14. Flow field of the pipe bends in different bending angle. Fig.15. Effect of R/D ratio on the predicted penetration rate (CFD runs 2, 28-34): (a) penetration rate of the
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whole pipe bends; (b) penetration rate of the elbows.
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Fig.16. The erosion location for the pipe bends as the Stokes number changes. Location A is the side wall of the
Fig.17. The relationship between β and the Stokes number.
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elbow, location B is the outermost side of the elbow close to the elbow outlet.
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Fig.18. Sample particle trajectories and velocity vectors for standard pipe bend.
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vp2 up1
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up2
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vp2
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α1
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vp1
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vp1
up2
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Fig.1. Definition of the impact procedure of a particle: (a) straight wall; (b) curved wall.
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Fig.2. Computational geometry used in the simulation.
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Fig.3.Computational mesh used in the simulation.
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Fig.4. Comparison of predicted velocity profiles with experimental data of Enayet et al.[41]
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Fig.5. Comparison of predicted and experimental penetration rate along elbow curvature angle for the five erosion
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models with different particle-wall rebound models.
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Fig.6.Comparison of measured erosion from Bourgoyne [28] with different erosion models.
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Fig.7. Effect of inlet velocity on predicted penetration rate (CFD runs 1-5): (a) penetration rate of the whole
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pipe bends; (b) penetration rate of the elbows.
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Fig.8. Effect of particle diameter on predicted penetration rate (CFD runs 2, 6-12): (a) penetration rate of the
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whole pipe bends; (b) penetration rate of the elbows.
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Fig.9. Erosion profiles and particle trajectories of pipe bend with different particle diameter:(a) 50μm； (b)150μm;(c)300μm.
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Fig.10. Effect of pipe diameter on predicted penetration rate (CFD runs 2, 13-17): (a) penetration rate of the
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whole pipe bends; (b) penetration rate of the elbows.
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Fig.11. Effect of particles mass flow rate on predicted penetration rate (CFD runs 2, 18-22): (a) penetration rate
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Fig.12. Effect of bend orientation on predicted penetration rate (CFD runs 2, 23-25): (a) penetration rate of the
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Fig.13. Effect of bending angle on predicted penetration rate (CFD runs 2, 26 and 27): (a) penetration rate of the
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whole pipe bends; (b) penetration rate of the elbows.
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Fig.14. Flow field of the pipe bends in different bending angle.
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Fig.15. Effect of R/D ratio on the predicted penetration rate (CFD runs 2, 28-34): (a) penetration rate of the
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Fig.16. The erosion location for the pipe bends as the Stokes number changes. Location A is the side wall of
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Fig.17. The relationship between β and the Stokes number.
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Fig.18. Sample particle trajectories and velocity vectors for standard pipe bend.
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ACCEPTED MANUSCRIPT Table captions
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Table 1 Values of parameters in solid particle erosion models.
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Table 2 Experimental conditions from Zeng et al. [29].
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Table 3 Experimental conditions from Bourgoyne[28].
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Table 4 Conditions of the CFD runs.
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ACCEPTED MANUSCRIPT Table 1 Values of parameters in solid particle erosion models.
b 22.7
α0 15
w 1
x 3.147
y 0.3609
DNV A1 9.370
A2 42.295
A3 110.864
A4 175.804
A5 170.137
A6 98.398
E/CRC R1 5.3983
R2 -10.1068
R3 10.9327
R4 -6.3283
R5 1.4234
Oka k0 65
k1 -0.12
k2 2.3(Hv)0.038
k3 0.19
n1 0.71(Hv)0.14
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MA D TE CE P AC
52
z 2.532
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Ahlert a -38.4
n2 2.4(Hv)-0.94
A7 31.211
A8 4.170
V’/(m/s) 104
d’/μm 326
ACCEPTED MANUSCRIPT Table 2 Experimental conditions from Zeng et al. [29].
Value
Fluid Velocity Particle diameter Particle density Particle mass flow rate Brinell hardness (BH)of pipe wall Pipe material density
Water 4m/s 450μm 2650kg/m3 0.235kg/s 160 7800kg/m3
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CE P
TE
D
MA
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IP
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Name
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ACCEPTED MANUSCRIPT Table 3 Experimental conditions from Bourgoyne[28].
Fluid
D(mm)
R/D
u0(m/s)
dp(μm)
ρp (kg/m3)
Mp (kg/s)
ρw (kg/m3)
BH
1 2 3
Water Water Water
52.5 52.5 52.5
1.5 3 3.25
9.45 11.49 14.63
350 350 350
2650 2650 2650
1.52 3.71 2.36
7800 7800 7800
120 140 140
AC
CE P
TE
D
MA
NU
SC R
IP
T
Case
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ACCEPTED MANUSCRIPT
4
Water
5
Water
6
Water
7
Water
8
Water
9
Water
10
Water
11
Water
12
Water
13
Water
14
Water
15
Water
16
Water
17
Water
18
Water
19
Water
20
Water
21
Water
ρp (kg/m3)
Mp (kg/s)
ρw (kg/m3)
BH
90
40
1.5
5
200
2650
0.2
7800
140
90
40
1.5
10
200
T
Water
dp(μm)
2650
0.2
7800
140
90
40
1.5
15
200
IP
3
u0(m/s)
2650
0.2
7800
140
90
40
1.5
20
200
2650
0.2
7800
140
90
40
1.5
25
200
2650
0.2
7800
140
90
40
1.5
10
50
2650
0.2
7800
140
90
40
1.5
10
60
2650
0.2
7800
140
90
40
1.5
10
80
2650
0.2
7800
140
90
40
1.5
10
100
2650
0.2
7800
140
40
1.5
10
150
2650
0.2
7800
140
90
40
1.5
10
250
2650
0.2
7800
140
90
40
1.5
10
300
2650
0.2
7800
140
90
100
1.5
10
200
2650
0.2
7800
140
90
200
1.5
10
200
2650
0.2
7800
140
90
300
1.5
10
200
2650
0.2
7800
140
90
400
1.5
10
200
2650
0.2
7800
140
90
500
1.5
10
200
2650
0.2
7800
140
90
40
1.5
10
200
2650
0.05
7800
140
90
40
1.5
10
200
2650
0.1
7800
140
90
40
1.5
10
200
2650
0.15
7800
140
90
40
1.5
10
200
2650
0.25
7800
140
90
SC R
Water
R/D
NU
2
H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward
D(mm)
MA
Water
Bending angle(°)
D
1
Orientation
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Fluid
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CFD run
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Table 4 Conditions of the CFD runs.
55
27
Water
28
Water
29
Water
30
Water
31
Water
32
Water
33
Water
34
Water
2650
0.3
7800
140
90
40
1.5
10
200
2650
0.2
7800
140
90 90
40 40
1.5 1.5
10 10
200 200
2650 2650
0.2 0.2
7800 7800
140 140
45
40
1.5
10
200
T
Water
200
2650
0.2
7800
140
180
40
1.5
10
200
IP
26
10
2650
0.2
7800
140
90
40
2
10
200
2650
0.2
7800
140
90
40
2.5
10
200
2650
0.2
7800
140
90
40
3
10
200
2650
0.2
7800
140
90
40
4
10
200
2650
0.2
7800
140
90
40
5
10
200
2650
0.2
7800
140
90
6
10
200
2650
0.2
7800
140
40
8
10
200
2650
0.2
7800
140
40
90
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Water Water
1.5
NU
24 25
40
MA
Water
90
TE
23
H-V downward V-H downward V-H upward H-V upward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward
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Water
AC
22
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ACCEPTED MANUSCRIPT
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ACCEPTED MANUSCRIPT
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Graphical Abstract
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Three types of collisions occur due to the change of Stokes numbers in liquid-solid flow, namely, secondary flow driven collision, direct collision and sliding collision.
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ACCEPTED MANUSCRIPT Highlights
The two-way coupled Eulerian-Lagrangian approach is employed to solve the liquid-solid flow
The relationship between Stokes number and the dynamic movement of the maximum
IP
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in the pipe bend.
Three collision mechanisms are proposed to explain how the changes of Stokes numbers
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D
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influence the erosion location.
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erosion location is presented
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S0032-5910(16)30071-7 doi: 10.1016/j.powtec.2016.02.030 PTEC 11509
To appear in:
Powder Technology
Received date: Revised date: Accepted date:
11 September 2015 12 February 2016 15 February 2016
Please cite this article as: Wenshan Peng, Xuewen Cao, Numerical prediction of solid particle erosion in pipe bends with liquid–solid flow, Powder Technology (2016), doi: 10.1016/j.powtec.2016.02.030
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Numerical prediction of solid particle erosion in pipe bends with
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Wenshan Peng, Xuewen Cao*
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liquid-solid flow
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College of Pipeline and Civil Engineering, China University of Petroleum, No.66, Changjiang West Road, Qingdao 266580, China
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*Correspondence: Prof. X. Cao ([email protected]), College of Pipeline and Civil Engineering,
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China University of Petroleum, No.66, Changjiang West Road, Qingdao, 266580, China
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Abstract
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Erosion caused by solid particles in pipe bends is one of the major concerns in the oil and gas industry which may result in equipment malfunctioning and even failure. In this work, a two-way
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coupled Eulerian-Lagrangian approach is employed to solve the liquid-solid flow in the pipe bend. Five different erosion models and two particle-wall rebound models are combined to predict the
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erosion rate. The most accurate model is chosen to calculate the effects of a range of parameters on erosion after comparing the predicted results with the experimental data. Further, the relationship between the Stokes number and the maximum erosion location is also assessed. It is found that although all these erosion models generate qualitatively similar erosion patterns, the Erosion/Corrosion Research Center (E/CRC) erosion model with the Grant and Tabakoff particle-wall rebound model produces results that are closest to the experimental data. The sequences of the most influencing factors are obtained: pipe diameter, inlet velocity, bending angle, particle mass flow, particle diameter, and Mean Curvature Radius/Pipe Diameter (R/D) ratio and bend orientation. Additionally, the relationship between Stokes number and the dynamic 1
ACCEPTED MANUSCRIPT movement of the maximum erosion location is presented which can be used to predict the maximum erosion location for different operating conditions. Three collision mechanisms are
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proposed to explain how the changes of Stokes numbers influence the erosion location.
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Keywords: Erosion prediction; Elbow; Two-way coupling; Liquid-solid; Stokes number
1. Introduction
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Erosion caused by solid particles can be considered as a severe problem in the oil and gas industry. Sand is always entrained in the transporting fluid produced from the well. The small
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solid particles flow with the carried fluid and impact the inner wall of the piping, valves and some
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other components. The components face a high risk of solid particle erosion due to the constant
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collision, which may result in equipment malfunctioning and even failure [1-4]. As a complex process, erosion is affected by lots of factors. Particularly in the pipelines, small or subtle change
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of operating conditions can influence the damage due to erosion significantly. Obtaining accurate erosion regularities of different influencing factors is essential for predicting the service life of
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pipelines. Additionally, the accurate erosion prediction can also help in finding the spots where severe erosion is more likely to occur [2]. Many erosion models of solid particles are proposed to calculate the erosion rate of different components. Meng and Ludema [5] carried out a detailed investigation of the erosion models developed previously, and found 28 erosion models were related to solid particle impingement, as well as 33 key parameters that affect erosion rate. Several erosion models mentioned in their work are popularly used in predicting erosion rate for pipe bends, such as the models proposed by Finnie [6], Bitter [7, 8], Neilson and Gilchrist [9], Grant and Tabakoff [10], and Hutchings [11]. The review indicated that each erosion model was developed based on a specific erosion 2
ACCEPTED MANUSCRIPT mechanism and no single predictive equation could be used for practical erosion prediction. Since the 1990s, computational fluid dynamics (CFD) has been widely used for predicting erosion
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caused by solid particles. The CFD method greatly promotes the development of the erosion
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models and several widely used CFD-based erosion models are proposed [12, 13]. Chen et al. [14]
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applied a CFD-based erosion prediction model that was developed by Ahlert [15] and McLaury [16] to predict the relative erosion severity between the elbow and the plugged tee with water/sand
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flow. The particle trajectories and the erosion pattern were analyzed by employing the Grant and Tabakoff [17] particle-wall rebound model. Wood and co-wokers [18] studied the particle
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distributions and particle impingement conditions in particle-laden liquids in Horizontal-vertical
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(H-V) upward pipe bends. The Hashish [19] erosion model was implemented into the CFD
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software to study the effect of impact angles and impact velocities on the pipe. They found that erosion always occurred in specific areas. In another study [20], they performed their erosion
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research on small curvature bend and an upstream straight pipe section. The fluid phase was modeled using slurry flow. The in-plane wall erosion rate calculated by the Hashish erosion model
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[19] was in good agreement with their experimental data. Huang et al. [21] proposed a phenomenological erosion model based on their previous study [22] to calculate the erosion rate of material in slurry flow. The paper suggested that the erosion rate showed a strong dependence on the slurry mean velocity and a weak dependence on pipe diameter and fluid viscosity. The erosion rate has a power-law relation with particle diameter, slurry mean velocity, pipe diameter and liquid viscosity. To investigate the effect of different factors on erosion of liquid-solid flow, various experimental methods have been adopted, such as the slurry pot test [23], jet impingement test [24], Coriolis erosion test [25] and pipe loop test [20,26-29]. The test methods except for the pipe 3
ACCEPTED MANUSCRIPT loop test can only study the erosion of pipe materials and cannot be employed to evaluate the erosion of the real pipelines. However, the loop erosion tests of elbows with liquid-solid flow are
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much fewer than the other tests due to their huge consumption of time and great complexity of
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monitoring. Blanchard et al. [27] studied the effect of Mean Curvature Radius/Pipe Diameter (R/D)
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ratio and particle size on the pipe erosion by using the circulating loop system. They found that the value of the maximum erosion angle was almost the same for elbows under the conditions of
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different size particles or elbow properties. Wood et al. [20] used an experimental loop to explore the erosion rate of the straight and curved ducts. By comparing the results of three different test
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methods, they found that there was a remarkable increase for the erosion of the outermost side
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wall of the bends compared to the innermost side wall and significant erosion occurred at the base
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of the bend. Bourgoyne [28] measured the erosion rate of pipe bend with liquid-solid flow. He studied the effect of R/D ratio, particle mass flow rate and particle velocity on the erosion rate and
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obtained the maximum erosion angle. Although his experimental data of liquid-solid erosion is few, it was referenced by many investigators as a data base to develop erosion equations. Zeng et
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al. [29] investigated the erosion-corrosion (E-C) behavior of an X65 pipeline elbow by using the array electrodes technique. He studied the percentages of the pure erosion rate and found that most of the erosion occurred at the outermost side of the elbow. Most of the currently available CFD-based erosion models and the experimental data of loop tests focus on the pipe bend with gas-solid flow. Studies on the erosion of the pipe bend with liquid-solid flow are relatively few and the accuracy of the erosion model in predicting the erosion of pipe with liquid-solid flow needs further validation. Since abrupt diversion will occur in the elbow section of the pipe which will lead to considerable difference in erosion, this work will look at particulate erosion of a 90° elbow in more detail. The Eulerian-Lagrangian approach is used to 4
ACCEPTED MANUSCRIPT solve the liquid-solid flow. The particle trajectory and the fluctuations of the liquid phase are simulated by the particle-eddy interaction method and the Discrete Radom Walk (DRW) model.
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Five erosion models and two particle-wall rebound models are combined to calculate the erosion
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rate. The particle-fluid interaction is accounted for by using two-way coupling. The erosion
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regularities of the pipe bends and elbows in different flow conditions are analyzed by using the most accurate erosion model. Furthermore, the relationship of Stokes number and the locations
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which are prone to erosion is also studied, and this relationship is explained by three different collision mechanisms.
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2. Numerical modelling
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The Eulerian-Lagrangian method is used in this study. The liquid is treated as a continuous
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phase and solved by the Navier-Stokes equations, while particles are treated as a discrete phase and solved by Newton’s second law. Erosion modeling based on the CFD consists of three steps:
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the continuous phase flow field simulation, particle tracking, and erosion calculation. The first two steps will be described in this section and the last step will be described in the next section.
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2.1. Liquid phase model
The Navier-Stokes equations are employed in this section. The continuity equations and momentum equations are written as:
( v ) 0 t
( v ) ( vv ) p ( ) g S M t
(1) (2)
where ρ is the liquid density, v is the instantaneous velocity vector, p is the pressure, is the stress tensor, g is the body force, S M is the added momentum due to the solid phase. The stress tensor is given by:
5
ACCEPTED MANUSCRIPT 2 3
[(v v T ) vI ]
(3)
where μ is the molecular viscosity, I is the unit tensor.
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The standard k-ε model is used to resolve the flow turbulence, and the equations are given as: (4)
( ) ( ui ) 2 [( t ) ] C1 Gk C2 S t xi x j x j k k
(5)
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( k ) ( kui ) k [( t ) ] Gk Sk t xi x j k x j
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where Gk is the generation of turbulence kinetic energy due to the mean velocity gradients, C1ε, C2ε are constants, xi and xj are the spatial coordinates, σk and σε are the turbulent Prandtl numbers
k2
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for k and ε, Sk and Sε are source terms, t C
, σk=1.0, σε=1.3, C1ε=1.44, C2ε=1.92, Cμ =0.09.
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2.2. Disperse phase model
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The particle trajectory is acquired by integrating the motion equation of the particles under the
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Lagrangian coordinates. While setting up the particle tracking and calculating the erosion rate, the following assumptions are made: (1) the injected particles are generally independent of each other
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and the reaction between particles is neglected, (2) the particle breakage is neglected, (3) the modifications of the elbow caused by the particles impaction are neglected. The governing equation of particle motion is proposed according to Newton’s second law: mp
dup FD FP FVM FB dt
(9)
where FD , FP , FVM ,and FB represent the drag force, the pressure gradient force, added mass force and buoyancy force, the expressions of which are given as: FD CD f
d p2 8
| u u p | (u u p )
1 FP d p3P 4 6
(10) (11)
ACCEPTED MANUSCRIPT 1 dup d p3 p 12 dt
(12)
1 FB d p3 ( p ) g 6
(13)
d p | u p u | , mp is the particle mass, up is the particle velocity vector, dp is the
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number, Re s
a a2 3 2 , Res is the particle Reynolds Re s Re s
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where CD is the drag coefficient， CD a1
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FVM
particle diameter, ρp is the density of particles, ρ is the density of fluid, μ is the viscosity of the
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fluid, a1, a2, a3 are constants for the smooth spherical particles given by Morsi and Alexander [30].
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2.3. Coupling between the two phases
The fluid carrier influences the dispersed phase via drag and turbulence, and the particles in
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turn influence the carrier fluid via the reduction in mean momentum and turbulence. The two-way
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coupling is used to solve the interaction between the particles and the liquid. The particle source
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terms are applied to Eqs. (2), (4) and (5) to take these effects into account. 2.3.1. Momentum coupling
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The momentum exchange is computed by examining the change of the particle momentum when it passes through each control volume. This momentum change is computed as:
SM (FD FP FVM FB ) M p t
(14)
where Mp is the mass flow rate of the particles,Δt is the time step. 2.3.2. Turbulence coupling The fluid velocity includes two elements in turbulent flow, the mean velocity and the random fluctuation velocity. The second term can influence particle trajectories. Various methods [31-33] have been proposed to calculate this term. In this paper, the particle-eddy interaction model proposed by Gosman and Ioannides [31] is used to calculate the effect of turbulent fluctuations on 7
ACCEPTED MANUSCRIPT particles. The effect is considered using the DRW model. The turbulent fluctuating velocity that follows a Gaussian distribution is given by:
u ' (u ' ) 2
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where ζ is a random number that obeys normal distribution.
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(15)
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Supposing the local turbulence is isotropic, the local root mean square value of the velocity fluctuation can be calculated by the local turbulent kinetic energy of the flow field: (16)
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( u ' ) 2 2k / 3
Two-way turbulence coupling enables the effect of the change in turbulent quantities due to
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particle damping and turbulence eddies. To take this effect into consideration, the particle source
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terms are included in the k-ε turbulence model, which are in Eqs. (4) and (5). The turbulence
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kinetic energy of liquid phase is modified by the formulation described in [34, 35]. 2.4. Particle impact and rebound behavior
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The particle-wall rebound model is essential in calculating the rebound angle and rebound velocity of particles after impacting the wall. It is also very important for simulating the particle
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trajectories when using the Lagrangian approach. Some commonly used models were developed by Grant and Tabakoff [17], Forder et al. [36], and Sommerfeld and Huber [37]. When a particle impacts the wall, it will lose energy. The rebound velocity is lower than the incident velocity. The restitution coefficient is incorporated to take this effect into consideration. en and et are the restitution coefficients for normal and tangential velocity components which show the change in particle velocity after collision. en and et are written as:
en
up2
et
v p2
u p1
v p1 8
(17)
(18)
ACCEPTED MANUSCRIPT where up1 and up2 are the normal components of the particle velocity before and after impact, vp1 and vp2 are the tangential component of the particle velocity before and after impact, as shown
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in Fig.1.
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In this work, the stochastic particle-wall rebound model by Grant and Tabakoff [17] and the
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non-stochastic particle-wall rebound model by Forder et al. [36] are used with the erosion prediction models to track particles and calculate the erosion. The model proposed by Forder et al.
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[36] is given by:
(19)
et 1 0.78 0.84 2 0.21 3 0.028 4 0.022 5
(20)
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en 0.988 0.78 0.19 2 0.024 3 0.027 4
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where α is the particle incidence angle.
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The model proposed by Grant and Tabakoff [17] is given by: (21)
et 0.988 1.66 2.11 2 0.67 3
(22)
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3. Erosion models
en 0.993 1.76 1.56 2 0.49 3
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Alternative methods of predicting erosion rates have been suggested by many investigators. The solid particle erosion models are in the general form:
ER Kf ( )u np
(23)
where ER is the erosion rate of the target. K is a constant depending on the target property, particle shape, particle hardness and some other factors, f(α) is a dimensionless function of the impact angle, n is a material dependent index. The erosion rate is defined as the wall mass loss per unit area and per unit time (kg/m2s).The erosion is finally given as the penetration rate (m/s), which is evaluated by dividing the erosion rate (kg/m2s) by the density of pipe wall material (kg/m3). As the penetration rate calculated in this study is very small, we use a smaller unit (nm/s) 9
ACCEPTED MANUSCRIPT to represent the erosion throughout this work. Five frequently used erosion models are analyzed and compared in this section.
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3.1. Ahlert erosion model
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Ahlert [15] developed an erosion model for AISI 1018 steel. This model was based on a series
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of direct impact experiments for different particle shapes and impact angles. The erosion model is given by:
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ER A( BH )0.59 Fsu np f ( )
f ( ) a 2 b
0
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f ( ) x cos2 sin( w ) y sin 2 z 0
(24) (25) (26)
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where A is a constant, A=15.59×10-7 for carbon steel, α is the particle impact angle, α0 is the
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transition angle, normally set as15°, n=1.73, a, b, w, x, y, z are listed in Table 1. Fs is the particle shape coefficient (Fs=1.0 for sharp particles, 0.53 for semi-rounded particles and 0.2 for fully
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rounded particles). BH is the Brinell’s Hardness of the target material. 3.2. DNV erosion model
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DNV [38] proposed an erosion model for various target materials based on a large number of experimental data. This model can predict the erosion for steel, epoxy, titanium, vinyl ester. As the model is relatively simple, it can be used in the CFD software easily.
ER Cf ( )u np
(27)
8
f ( ) ( 1)i 1 Ai i
(28)
i 1
where C is a constant determined by the target material. For steel pipes, C=2.0×10-9, n is the velocity exponent, for steel pipes, n=2.6. Values of Ai are given in Table 1.
10
ACCEPTED MANUSCRIPT 3.3. E/CRC erosion model The Erosion/Corrosion Research Center (E/CRC) at the University of Tulsa [13] developed the
given according to Zhang et al. [4] as:
5
f ( ) Ri i i 1
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ER C ( BH )0.59 FS u np f ( )
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E/CRC models to predict the erosion of elbows, tees and some other pipe fittings. The model is
(29) (30)
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where C is the constant and equal to 2.17×10-7 for carbon steel, n=2.41. Values of Ri are listed
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in Table 1.This model is similar to the Ahlert erosion model, but the constant C is much smaller than A. The E/CRC erosion model is a little more complicated than the DNV erosion model
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because the particle hardness and the particle sharpness are added into consideration.
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3.4. Neilson and Gilchrist erosion model
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Neilson and Gilchrist [9] proposed an erosion model based on the impact angle in the basis of Finnie and Bitter’s study. They developed two equations for particles erosion of large impact
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angles and small impact angles and suggested that the total erosion rate was equal to the summation of erosion due to deformation and cutting mechanisms. This model was derived based on the experimental results:
ER
ER
u 2p cos2 sin 2 C u 2p cos2 2 C
20
u 2p sin 2 2 D
u 2p sin 2 2 D
0
(31)
0
(32)
where α0 is the transition angle, normally set as 45°, εC is the cutting coefficient, set as 3.332× 107, εD is the deformation coefficient, set as 7.742×107.
11
ACCEPTED MANUSCRIPT 3.5. Oka et al. erosion model Oka et al. [39, 40] developed an erosion model by taking into account many parameters that
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affect the material erosion. An angle function combined with two terms was proposed to calculate
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the erosion damage. This model considers more influencing factors than the above models, such as
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the particle diameter, the reference diameter and the impact speed of the reference particle, thus it has a more broad application. This erosion model is expressed as:
up
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ER 1.0 109 wk0 f ( )( Hv )k1 (
V
'
) k2 (
dp d
) k3
(33) (34)
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f ( ) (sin )n1 [1 Hv(1 sin )]n2
'
where ρw is the density of target material, Hv is the Vickers hardness of the target material, dp
D
is the particle diameter, d’ is the reference diameter, V’ is the impact speed of the reference particle,
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k0, k1, k2, k3, n1, n2, d’, V’ are listed in Table 1.
4.1. Case description
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4. CFD modeling
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The commercial software ANSYS FLUENT is the CFD solver used for the numerical simulations. A database of the experiment conducted by Zeng et al. [29] is employed in this work to investigate the performed erosion models. Zeng et al. studied the erosion of an elbow using a circulating loop system. 38 sets of experimental data were gathered on the same elbow. The test piece was a 90°elbow with a diameter of 50 mm and a curvature radius of 76.9 mm, which can be seen in Fig.2. In order to better represent the flow at the specimen location, a 1m (20D) vertical pipe upstream and a 0.5m (10D) horizontal pipe downstream of the elbow were used. The flow conditions are shown in Table 2. 4.2 Computational mesh
Fig.3 shows the 3-D computational domain used in this work. Generally, meshing consists of 12
ACCEPTED MANUSCRIPT two parts: surface meshing and volume meshing. The surface grids are carefully generated due to its important effect on the quality of resulting volume grids. In order to obtain an accurate flow
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field near the pipe wall, a finer grid scheme in the boundary layer must be used. The boundary
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layer near the pipe inner wall has five-layer grids (the first percent of the cell height of the first
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layer is 20% and the growth factor is 1.2). The structured grid is used to mesh the surface of the cross-section. Finally, the hexahedral structured mesh is adopted to mesh the whole volume. In
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present study, grid independency study has shown that larger number of grids will not necessarily influence the accuracy of the cases when the grid quantity achieves a certain number. The grid
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number used in the base case is approximately 850,000. 4.3. Boundary conditions
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In this work, the solid particles are injected uniformly at the pipe inlet at the same speed as
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the water. The working pressure is set as 1.01×105 Pa and the temperature is set as 25 ℃. The
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turbulence intensity is set at 3.5%. Additionally, wall roughness height is set as 10μm which is consistent with the study of Zeng et al. [29]. The roughness constant is set as the default value of
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0.5. The wall boundaries are set as ‘reflect’ and the outlet boundary is set as ‘escape’. 4.4. The simulation procedure
The SIMPLEC algorithm is used to solve the pressure and velocity field in order to improve convergence. The standard discretization schemes are used for the pressure terms and the second order upwind discretization schemes are used for the convection terms and divergence terms. The convergent criteria for all the steady simulations are set as that the residual in the control volume for each equation is smaller than 10-5 or the number of iterations reaches to 5000. The number of continuous phase iterations per discrete phase model (DPM) iteration is set as 5. A total of 10260 particles will be tracked in the simulation. The simulations are executed in the Intel (R) Xeon (R) 13
ACCEPTED MANUSCRIPT with specifications of CPU E5-2690 2.90GHz and 16.0 GB RAM with windows server 2008 platform. About 6 CPU hours are consumed for one case.
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5. Results and discussion
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5.1. Flow model verification
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Since the flow field simulation is the first step of the erosion calculation, it is very important to the accuracy of the erosion predictions. In order to verify the flow modeling capabilities, a
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verification study is performed on a 90°elbow, as shown in Fig.4. By comparing the predicted velocity profiles with the experimental data of Enayet et al. [41], the accuracy of the flow solution
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can be assessed. Enayet et al. studied the flow field of a 90° pipe bend with the R/D ratio equal to 2.8 using the Laser Doppler Velocimetry (LDV) technique. They obtained the velocity profiles of
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different sections of an elbow for a turbulent flow case with Reynolds number equal to 43000. Measurements were made in planes perpendicular to the flow direction at 30°,60° and 75°。
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Comparison of predicted velocity profiles and experimental velocity profiles is shown in Fig.4. There is good agreement between the predictions and the experimental data. It is worth noticing
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that the k-ε model can successfully predict the changes of flow in the pipe bend. The velocity changes more dramatically near the inside wall than that near the outside wall. This may be attributed to the effect of the secondary flow at the elbow. These secondary flows can significantly influence the erosion profiles. The relationship between erosion profiles and the secondary flow are described in the following sections. 5.2. Comparison among erosion models Fig.5 shows the penetration rate around the elbow obtained by five CFD-based erosion models and the experimental data by Zeng et al. [29]. It can be noticed that the Ahlert erosion model results in much higher penetration rate compared to the experimental data. The penetration rate 14
ACCEPTED MANUSCRIPT calculated by Oka et al. erosion model and Neilson and Gilchrist erosion model also over-predict the penetration rate. The reason may be that the flow conditions and the particle property used in
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Zeng’s experiment are not similar to the conditions in the numerical models. Specificity of the
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material used in Neilson and Gilchrist experiment may be also a contributory factor to the
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over-prediction. Fig.5 shows that both the predictions obtained by the DNV erosion model and by the E/CRC erosion model show good agreement with the experimental data. As the erosion data
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from Zeng et al. [29] was only about the erosion around the elbow under a certain experimental condition, more experimental data is needed to further verify which model can produce results that
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are closest to the experimental data. Bourgoyne [28] studied solid particle erosion in diverter
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systems and several experimental data were gathered for sand erosion in liquid flow. Although
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only several data are available, they can help us find the most accurate erosion model after comparing the numerical erosion rate to the experimental data. The flow conditions of the
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experiments conducted by Bourgoyne[28] are described in Table 3. As shown in Fig.6, the ratios of predicted penetration to experimental penetration lie in the
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range of 1.01-1.82 when the E/CRC model and the Grant and Tabakoff model are applied, which are smaller than the other three sets of predictions. It indicates that the prediction calculated by E/CRC erosion model with the Grant and Tabakoff particle-wall rebound model yields the best agreement with the experimental data. Thus, in the following sections, the effect of various influencing factors such as inlet velocity, particle diameter, pipe diameter, particle mass flow rate, bend orientation, bending angle and R/D ratio will be assessed by using this model. The CFD simulations are performed under different flow conditions as described in Table 4.
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ACCEPTED MANUSCRIPT 5.3. Effect of inlet velocity on predicted penetration rate Fig.7 shows the effect of inlet velocity on predicted penetration rate. Fig.7a indicates that the
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penetration rate increases exponentially as the inlet velocity increases which means that the
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particle velocity has a great influence on the penetration rate. A larger inlet velocity will result in
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larger impact energy and as a result, the penetration rate is higher. Significant difference of the first erosion peak is observed in Fig.5b, the maximum erosion under low inlet velocity appears
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earlier than that under high inlet velocity. For inlet velocity of 5m/s, the first remarkable erosion peak appears at 67°.However, it rises to 86° for inlet velocity of 25m/s. Fig.7b also displays that
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under the same inlet velocity, the penetration rate increases as the angle of the elbow increases.
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The maximum penetration location appears at 90° of the elbow.
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5.4. Effect of particle diameter on predicted penetration rate Fig.8 shows the effect of particle diameter on predicted penetration rate. Fig.8a suggests that
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as the particle diameter increases, the penetration rate first decreases then increases, with the minimum value occurring at 150μm. Thus, 150μm is the critical diameter for the erosion. The
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redirection of the liquid in the elbows has great influence on the particles, especially for the small particles. Fig.8a presented that the penetration rate in the condition of 50μm is even higher than that of 200μm. The reason for this special phenomenon may be that the dominant force for the particles with different diameter is changed. For small particles, the intense secondary flow in the elbows causes the particles to impact the side wall of the elbow (region p1) and results in severe erosion (region s1), as shown in Fig.9. For large diameter particles, the significance of the secondary flow effect on the particles decreases with respect to the inertia force. The inertia force drives the particles impact directly on the elbow (region p3) and cause severe erosion in the outside wall (region s3). For the medium diameter particles, both the secondary flow effect and the 16
ACCEPTED MANUSCRIPT inertia force are remarkable, these two mechanisms separate the particles into two major paths (region p2) and cause a large scale erosion scar on the inner surface of the elbow (region s2). Since
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the particles are separated into two parts by the two mechanisms, the penetration rate maybe the
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lowest compared to the other two conditions.
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It can also indicate that as the particle diameter increases, the maximum penetration rate will occur at a lower angle, as depicted in Fig.8b. This is because the trajectories of large particles are
then impact the outer wall at a lower angle.
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mainly controlled by inertia force. This force will drive the large particles flow in straight path and
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5.5. Effect of pipe diameter on predicted penetration rate
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Fig.10 shows the effect of pipe diameter on predicted penetration rate. In Fig.10a, the
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penetration rate decreases sharply as the pipe diameter increases. The penetration rate decreases from 0.681nm/s to 0.0122nm/s as the pipe diameter increases from 40mm to 400mm, which
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reduces more than 50 times. When the pipe diameter exceeds 400mm, the penetration rate will decrease slowly and stabilize at a very low level. This effect may be attributed to the fact that the
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pipe with large diameter has a relatively larger impact surface. Since the particle mass flow rate is kept at 0.02kg/s for these cases, the erosion rate can be defined as the penetration rate per unit area (nm/s/m2) to make more sense for this comparison. For example, the value of erosion rate is 5.81×10-9 nm/s/m2 for the elbow with a diameter of 400mm and 2.78×10-9 nm/s/m2 for the elbow with a diameter of 500mm. The discrepancy between them is not remarkable. Moreover, the maximum penetration appears at between 75° and 90° for all pipe diameters in Fig.10b. 5.6. Effect of particles mass flow rate on predicted penetration rate Fig.11 shows the effect of particles mass flow rate on predicted penetration rate. The 17
ACCEPTED MANUSCRIPT penetration rate increases linearly as the particle mass flow increases, as shown in Fig.11a. This can be explained that as the mass flow rate increases, more solid particles will impact the pipe
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wall at a time and result in higher penetration rate. Under the same particle mass flow rate, the
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penetration rate increases slowly with the increasing of the degree, as shown in Fig.11b. Zheng et
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al. [42] examined the effect of particle mass flow rate and found that the penetration rate will no longer increase when the mass flow rate increases to a certain value. As the mass flow rate
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increases, the loss of the momentum from the collisions between particles becomes dramatically. Thus, the effect of the particle mass flow decreases at high particle mass flow rates. This effect is
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not shown in our results, however. This may be attributed to the fact that we use the two-way
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coupling in this work and the particle-particle interactions are neglected. Further under-predictions
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are anticipated if the effect of particle-particle interactions is taken into consideration in the numerical simulations. For high loading flow, the four-way coupling which considers the
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particle-particle interactions will be necessary. 5.7. Effect of bend orientation on predicted penetration rate
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Fig.12 shows the effect of bend orientation on predicted penetration rate. It suggests that there is no significant difference in the penetration rate when the bend orientation changes. The erosion regularities of the elbows with different bend orientation are similar. The maximum erosion locations for these elbows all occur at about 85°. 5.8. Effect of bending angle on predicted penetration rate Fig.13 shows the effect of bending angle on predicted penetration rate. The penetration rate increases as the bending angle increases, as shown in Fig.13a.On one hand, due to geometrical shape of these three elbows, the impact angle of the particles at the elbow wall of the 90°elbow and 180°elbow is larger than that of the 45°elbow. The maximum impact angle for 90°elbow 18
ACCEPTED MANUSCRIPT and 180°elbow is 60°and changes to 45° for 45°elbow. When the impact angle increases from 45° to 60°, the value of f(α) in Eq.(30) first increases from 1.316 to 1.325 then decreases to 1.296. The
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average value of f(α) in the range of 45°to 60°is larger than the value of f(α) at 45°.Thus, the value
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of f(α) in large bending angle elbow is larger than that in small bending angle elbow. On the other
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hand, as the maximum velocity area of the large bending angle pipe bend is larger than that of small bending angle pipe bend in the elbow region, the particles in large bending angle pipe bend
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will get a larger impact velocity after crossing through this region, as shown in Fig.14. Since f(α) and particle impact velocity directly increase the erosion rate of pipe bend which can be seen in
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Eqs.(29) and (30), the large bending angle elbow experiences more severe erosion than small
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bending angle elbow.
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Additionally, the particles in larger angle elbows experience multiple collisions with the inner wall due to the fact that the length of large angle elbow is greater than that of small angle elbow.
shown in Fig13b.
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As a result, the 180° elbow presents more penetration peaks than 90° elbow and 45° elbow, as
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5.9. Effect of R/D ratio on predicted penetration rate Fig.15 shows the effect of R/D ratio on predicted penetration rate. The penetration rate decreases slowly as the R/D range from 1.5 to 4, and then levels off slightly as the R/D ratio is greater than 4, as shown in Fig.15a. Therefore, there is an economical ratio in the engineering practice. Since the diameters of the pipe bends are set as 40mm in these runs, the larger R/D ratio makes the pipe path longer and as a result, the solid particles flow more smoothly and cause less impact on the wall. The secondary flow influences the penetration rate more significantly as the R/D ratio increases. The secondary flow will occur earlier and drive the particles to impact the side wall of the elbow. As a consequence, the erosion at the outermost of the elbow becomes less 19
ACCEPTED MANUSCRIPT obvious as shown in Fig.15b. Thus, it is beneficial to use long-radius elbows in the engineering practice.
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5.10. The relationship between Stokes number and the erosion locations
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The movement of particles through elbows is governed by three forces: particle inertia force,
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drag force exerted by surrounding water and secondary flow. The particle inertia force maintains the particle move in the tangential direction. The drag force motivates particles flow along the
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water streamline, and the secondary flow forcing solid particles move towards the outer wall. Stokes number as the ratio of particle relaxation time and fluid characteristic time can reflect the
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magnitude of the particle inertia force and the drag force and it is a dimensionless number for
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representing the curvilinear movement of the solid particle. Thus, Stokes number is adopted to
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The form of Stokes number is:
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evaluate the relationship between these three mechanisms and their effect on the pipe bend erosion.
St
p d p2u 18 D
(35)
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In order to briefly describe the severity of the erosion in elbows, the ‘erosion degree’ is imported in this section. The erosion degree includes three levels. The high erosion degree represents that the elbow experiences severe erosion and the low erosion degree represents that the elbow experience slight erosion, and the medium erosion degree is between them. It can be seen from Fig.16 that there are two dynamic severe erosion locations which are location A and location B respectively. The maximum erosion location will dynamic change as the Stokes number increases. Location A is exposed to greater erosion when the Stokes number is small. However, as Stokes number increases, this maximum erosion area will move to location B. When Stokes number is small, the drag force is dominant. The secondary flow plays an important role in the 20
ACCEPTED MANUSCRIPT movement of particles. The secondary flow first drives the particles in the circumferential direction from the outer side to the inner side of the elbow and then drives the particles to the side
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wall of the elbow and as a result, causing severe erosion in this area. For a larger Stokes number,
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the inertia force plays a larger role on the particles movement. The particles have enough
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momentum to flow across eddies, thus the fluid velocity and the fluid flow direction have a smaller influence on the particles. As a result, the solid particles deviate from the fluid streamlines
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and impact on the outer wall.
A dimensionless number is defined to analyze the effect of Stokes number on the maximum
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erosion rate of location B. This number describes the relative magnitudes of the maximum erosion rate of this area and the maximum erosion rate of the whole pipe bend, the number is given as:
Max.ERelbow Max.ER pipe
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(36)
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Fig.17 shows the relationship between β and the Stokes number. As the Stokes number increases, the number β first increases and then stabilizes at 1.0. The number β will no longer
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change when Stokes number is larger than 1.472. This indicated that 1.472 is a division value for Location A and Location B. When the Stokes number is smaller than 1.472, the maximum erosion will occur at Location A, otherwise, the maximum erosion will occur at Location B. The sample particle trajectories and velocity vectors for standard pipe bend are shown in Fig.18.There are three paths of particles, namely a-b-c-d, e-f-g and h-i-j. These three paths are controlled by inertia force, drag force and secondary flow, that is, they are influenced by the Stokes number. Location A can be explained by path a-b-c-d, as well as Location B can be explained by path e-f-g and h-i-j. For small Stokes numbers, the particles readily flow along the streamlines, thus these particles will change their flow direction due to the secondary flow which 21
ACCEPTED MANUSCRIPT first appears at location a, as shown in Fig.18. The secondary flow drives these particles to leave from location a and collide at location b. For the fluid having the largest velocity in the inner side
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of the elbow, the velocity of the particles become larger from location a to location b. The particles
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impact the inner side wall (Location A) with great kinetic energy and then cause severe erosion.
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The probability of rebounding to location c or d after colliding at location b depends on the particle impact energy, the flow field of the bend, the wall roughness, and the particle dispersion.
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In Fig.18, the erosion rate at location c and d are much lower than that at location a and location b. This is mainly because the flow field at downstream of the pipe bend is much smoother than that
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at the elbow. As the particle paths are almost parallel to the wall surface, the particle-wall impacts
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are at a much lower impact angle and the impact velocity perpendicular to the surface is also small.
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Both these effects decrease the erosion.
For larger Stokes numbers, the particles will move more independently and impact the
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outermost wall of the elbow directly which performs as path e-f-g and path h-i-j. For the particles with larger momentum, they will impact location e directly and as a result, the larger impact
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energy leads to heavy erosion at this location. Furthermore, after rebounding from location e, the particles will collide at location f and location g. However, the erosion rate at location f and location g is much lower than that at location e due to the smaller particle impact angle and the smaller impact velocity perpendicular to the surface. For particles in smaller velocity, the particles will flow from location h, and experience a sliding collision to location i, finally rebound to location j at a small rebound angle. The direct collisions and the sliding collisions determined these two paths and the erosion degree of location B. 6. Conclusions An Eulerian simulation with the k-ε turbulence model was used to simulate the liquid flow in a 22
ACCEPTED MANUSCRIPT 90° pipe bend, while the Lagrangian particle tracking method was employed for calculating the particle motion. The liquid-solid interaction was also taken into account. Five erosion models with
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two particle rebound models were combined to predict the erosion rate and the maximum erosion
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location. Based on the simulation results, it was concluded that the combination of the E/CRC
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erosion model with Grant and Tabakoff particle-wall rebound model is the most suitable combination for the erosion prediction because it provided the best agreement with the
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experimental data of Bourgoyne[28] and Zeng et al. [29].
The effects of different parameters on penetration rate were studied. The penetration rate
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increases as the inlet water velocity, mass flow rate and bending angle increase. On the other hand,
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the penetration rate decreases as the pipe diameter and R/D ratio increase. The bend orientation
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and flow direction have little effect on the change of the penetration rate. Particularly, it can be concluded that the penetration rate first decreases then increases, with the minimum value
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occurring at the particle diameter of 150μm. The penetration rate of small particles which are smaller than 150μm can even result in higher penetration rate than that of larger particles. Since in
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the CFD runs, only one influencing parameter is different at a time, the effect of that single parameter can be determined. The sequences of the most influencing parameters are: pipe diameter, inlet velocity, bending angle, particle mass flow, particle diameter, and the R/D ratio and bend orientation. Furthermore, the relationship between Stokes number and the dynamic movement of the maximum erosion location is presented. Erosion mainly occurs at the outermost side of the elbow and the side walls of the downstream straight pipe close to the elbow outlet. The most severe erosion location changes as Stokes number varies. For smaller Stokes numbers the maximum erosion location always occurs at the inner side walls of the elbow, while for larger Stokes 23
ACCEPTED MANUSCRIPT numbers, this location always occurs at the outermost side of the elbow. The Stokes number also influences the particle trajectories, thus, three particle paths are investigated to explain the erosion
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location. The results indicates that the erosion of the side wall is mainly caused by the secondary
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flow driven collision, while the erosion of outermost side of the wall is caused by a combined
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effect of direct collision and sliding collision. Acknowledgments
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The authors would like to acknowledge the financial support of National Natural Science Foundation of China (NO.51274232) and the Fundamental Research Funds for the Central
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Universities of China (No.15CX06070A).
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References
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[1] C. B. Solnordal, C. Y. Wong, J. Boulanger, An experimental and numerical analysis of erosion caused by sand pneumatically conveyed through a standard pipe elbow. Wear 336 (2015) 43-57.
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[2] G. C. Pereira, F. J. de Souza, D. A. de Moro Martins, Numerical prediction of the erosion due to particles in elbows. Powder Technol. 261 (2014) 105-117.
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[3] M. Jafari , Z. Mansoori, M. S. Avval, G. Ahmadi, A. Ebadi, Modeling and numerical investigation of erosion rate for turbulent two-phase gas–solid flow in horizontal pipes. Powder Technol. 267(2014)362-370. [4] Y. Zhang, E. P. Reuterfors, B. S. McLaury, S. A. Shirazi, E. F. Rybicki, Comparison of computed and measured particle velocities and erosion in water and air flows. Wear 263 (2007) 330-338. [5] H. C. Meng, K. C. Ludema, Wear models and predictive equations – Their form and content. Wear 181–183 (1995) 443–457. [6] I. Finnie, Erosion of surfaces by solid particles. Wear 3 (1960) 87–103. [7] J. Bitter, A study of erosion phenomena part I. Wear 6 (1963) 5-21. [8] J. Bitter, A study of erosion phenomena: part II. Wear 6 (1963) 169-190. 24
ACCEPTED MANUSCRIPT [9] J. H. Neilson, A. Gilchrist, Erosion by a stream of solid particles. Wear 11 (1968) 111-122. [10] G. Grant, W. Tabakoff. An experimental investigation of the erosive characteristics of 2024 aluminum alloy. No.
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73-37. Cincinnati: Department of Aerospace Engineering, University of Cincinnati, 1973.
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[11] I. M. Hutchings, R. E. Winter, J. E. Field. Solid particle erosion of metals: the removal of surface material by
Sciences. Vol. 348. No. 1654. The Royal Society, 1976.
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spherical projectiles. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering
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[12] D. O. Njobuenwu, M. Fairweather. Modelling of pipe bend erosion by dilute particle suspensions. Comput. Chem. Eng. 42 (2012)235-247.
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[13] M. Parsi, K. Najmi, F. Najafifard, S. Hassani, B. S. Mclaury, S. A. Shirazi, A comprehensive review of solid
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particle erosion modeling for oil and gas wells and pipelines applications. J. Nat. Gas Sci. and Eng. 21
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(2014) 850–873.
[14] X.Chen, B. S.McLaury, S. A. Shirazi, Numerical and experimental investigation of the relative erosion
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severity between plugged tees and elbows in dilute gas/solid two-phase flow. Wear 261 (2006) 715-729. [15] K. R. Ahlert, Effects of particle impingement angle and surface wetting on solid particle erosion of AISI 1018
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steel. Ph.D. Thesis, Department of Mechanical Engineering, The University of Tulsa, 1994. [16] B. S. McLaury, Predicting solid particle erosion resulting from turbulent fluctuations in oilfield geometries, Ph.D. Dissertation, The University of Tulsa, OK, 1996. [17] G. Grant, W. Tabakoff, Erosion prediction in turbomachinery resulting from environmental solid particles. J. Aircraft 12(1975)471-478. [18] R. J. K. Wood, T. F. Jones, N. J. Miles, J. Ganeshalingam, Upstream swirl-induction for reduction of erosion damage from slurries in pipeline bends. Wear 250 (2001) 770-778. [19] M. Hashish, An improved model of erosion by solid particle impact, in: Proceedings of the 7th International Conference on Erosion by Liquid and Solid Particle, Cambridge, 1987, pp. 66/1–66/9. 25
ACCEPTED MANUSCRIPT [20] R. J. K. Wood, T. F. Jones, J. Ganeshalingam, N. J. Miles, Comparison of predicted and experimental erosion estimates in slurry ducts. Wear 256 (2004) 937-947.
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[21] C. Huang, P. Minev, J. Luo, K. Nandakumar, A phenomenological model for erosion of material in a horizontal
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slurry pipeline flow. Wear 269 (2010) 190-196.
of materials in jet flow. Powder Technol. 187 (2008) 273-279.
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[22] C. Huang, S. Chiovelli, P. Minev, J. Luo, K. Nandakumar, A comprehensive phenomenological model for erosion
solid–liquid mixture. Wear 259 (2005) 196-202.
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[23] G. R. Desale, B. K. Gandhi, S. C. Jain, Improvement in the design of a pot tester to simulate erosion wear due to
MA
[24] E. Mahdi, A. Rauf, E. O. Eltai, Effect of temperature and erosion on pitting corrosion of X100 steel in aqueous
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silica slurries containing bicarbonate and chloride content. Corros. Sci. 83 (2014) 48-58.
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[25] Y. Xie, H. M. Clark, H. M. Hawthorne, Modelling slurry particle dynamics in the Coriolis erosion tester. Wear 225 (1999) 405-416.
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[26] C. A. Shook, M. McKibben, M. Small, Experimental investigation of some hydrodynamic factors affecting slurry pipeline wall erosion. Can. J. Chem. Eng. 68 (1990) 17-23.
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[27] D. J. Blanchard, P. Griffith, E. Rabinowicz, Erosion of a pipe bend by solid particles entrained in water. J. Manuf. Sci. Eng. 106 (1984) 213-217. [28] A. T. Bourgoyne Jr, Experimental study of erosion in diverter systems due to sand production. In SPE/IADC Drilling Conference. Society of Petroleum Engineers,1989. [29] L. Zeng, G. A. Zhang, X. P. Guo, Erosion–corrosion at different locations of X65 carbon steel elbow. Corros. Sci. 85 (2014) 318-330. [30] S. Morsi, A. J. Alexander, An investigation of particle trajectories in two-phase flow systems. J. Fluid. Mech. 55 (1972) 193-208. [31] A. D. Gosman, E. Ioannides, Aspects of computer simulation of liquid-fueled combustors. J. Energ.7 (1983) 26
ACCEPTED MANUSCRIPT 482-490. [32] G. Klose, B. Rembold, R. Koch, S. Wittig, Comparison of state-of-the-art droplet turbulence interaction models
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for jet engine combustor conditions. Int. J. Heat. Fluid. Fl. 22 (2001) 343-349.
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[33] C. Marchioli, A. Soldati, Mechanisms for particle transfer and segregation in a turbulent boundary layer.
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J. Fluid. Mech. 468 (2002) 283-315.
[34] A. A. Amsden, P. J. O'rourke, T. D. Butler, KIVA-II: A computer program for chemically reactive flows with
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sprays (No. LA-11560-MS). Los Alamos National Lab., NM (USA), 1989.
[35] G. M. Faeth, Spray atomization and combustion.Technical Report 1986-136, AIAA,1986.
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control valves. Wear 216 (1998) 184-193.
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[36] A. Forder, M. Thew, D. Harrison, A numerical investigation of solid particle erosion experienced within oilfield
Flow 25 (1999) 1457-1489.
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[37] M. Sommerfeld, N. Huber, Experimental analysis and modelling of particle-wall collisions. Int. J. Multiphase.
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[38] D. N. Veritas, Recommended Practice, Erosion Wear of Piping Systems. DNV RP O, 501, 2007. [39] Y. I. Oka, K. Okamura, T., Yoshida Practical estimation of erosion damage caused by solid particle impact. Part 1:
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Effects of impact parameters on a predictive equation. Wear 259 (2005) 95–101. [40] Y. I. Oka, T. Yoshida, Practical estimation of erosion damage caused by solid particle impact. Part 2: Mechanical properties of materials directly associated with erosion damage. Wear 259 (2005) 102–109. [41] M. M. Enayet, M. M. Gibson, A. M. K. P. Taylor, M. Yianneskis, Laser-Doppler measurements of laminar and turbulent flow in a pipe bend. Int. J. Heat. Fluid. Fl. 3 (1982) 213-219. [42] Y. G. Zheng, H. Yu, S. L. Jiang, Z. M. Yao, Effect of the sea mud on erosion-corrosion behaviors of carbon steel and low alloy steel in 2.4% NaCl solution. Wear 264 (2008) 1051-1058.
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ACCEPTED MANUSCRIPT Table captions Table 1 Values of parameters in solid particle erosion models.
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Table 2 Experimental conditions from Zeng et al. [29].
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Table 3 Experimental conditions from Bourgoyne[28].
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Table 4 Conditions of the CFD runs.
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Figure captions
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Fig.1. Definition of the impact procedure of a particle: (a) straight wall; (b) curved wall.
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Fig.2. Computational geometry used in the simulation.
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Fig. 3. Computational mesh used in the simulation.
Fig.4. Comparison of predicted velocity profiles with experimental data of Enayet et al.[41].
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Fig.5. Comparison of predicted and experimental erosion rate along elbow curvature angle for the five erosion models with different particle-wall rebound models.
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Fig.6. Comparison of measured erosion from Bourgoyne [28] with different erosion models.
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Fig.7. Effect of inlet velocity on predicted penetration rate (CFD runs 1-5): (a) penetration rate of the whole pipe
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bends; (b) penetration rate of the elbows.
Fig.8. Effect of particle diameter on predicted penetration rate (CFD runs 2, 6-12): (a) penetration rate of the
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whole pipe bends; (b) penetration rate of the elbows. Fig.9. Erosion profiles and particle trajectories of pipe bend with different particle diameter:(a) 50μm；
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(b)150μm;(c)300μm.
Fig.10. Effect of pipe diameter on predicted penetration rate (CFD runs 2, 13-17): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. Fig.11. Effect of particles mass flow rate on predicted penetration rate (CFD runs 2, 18-22): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. Fig.12. Effect of bend orientation on predicted penetration rate (CFD runs 2, 23-25): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. Fig.13. Effect of bending angle on predicted penetration rate (CFD runs 2, 26 and 27): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. 29
ACCEPTED MANUSCRIPT Fig.14. Flow field of the pipe bends in different bending angle. Fig.15. Effect of R/D ratio on the predicted penetration rate (CFD runs 2, 28-34): (a) penetration rate of the
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whole pipe bends; (b) penetration rate of the elbows.
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Fig.16. The erosion location for the pipe bends as the Stokes number changes. Location A is the side wall of the
Fig.17. The relationship between β and the Stokes number.
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elbow, location B is the outermost side of the elbow close to the elbow outlet.
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Fig.18. Sample particle trajectories and velocity vectors for standard pipe bend.
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ACCEPTED MANUSCRIPT Figure captions
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Fig.1. Definition of the impact procedure of a particle: (a) straight wall; (b) curved wall.
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Fig.2. Computational geometry used in the simulation.
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Fig. 3. Computational mesh used in the simulation.
Fig.4. Comparison of predicted velocity profiles with experimental data of Enayet et al.[41].
NU
Fig.5. Comparison of predicted and experimental erosion rate along elbow curvature angle for the five erosion models with different particle-wall rebound models.
MA
Fig.6. Comparison of measured erosion from Bourgoyne [28] with different erosion models. Fig.7. Effect of inlet velocity on predicted penetration rate (CFD runs 1-5): (a) penetration rate of the whole
TE
D
pipe bends; (b) penetration rate of the elbows. Fig.8. Effect of particle diameter on predicted penetration rate (CFD runs 2, 6-12): (a) penetration rate of the
CE P
whole pipe bends; (b) penetration rate of the elbows. Fig.9. Erosion profiles and particle trajectories of pipe bend with different particle diameter:(a) 50μm；
AC
(b)150μm;(c)300μm.
Fig.10. Effect of pipe diameter on predicted penetration rate (CFD runs 2, 13-17): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. Fig.11. Effect of particles mass flow rate on predicted penetration rate (CFD runs 2, 18-22): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. Fig.12. Effect of bend orientation on predicted penetration rate (CFD runs 2, 23-25): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. Fig.13. Effect of bending angle on predicted penetration rate (CFD runs 2, 26 and 27): (a) penetration rate of the whole pipe bends; (b) penetration rate of the elbows. 31
ACCEPTED MANUSCRIPT Fig.14. Flow field of the pipe bends in different bending angle. Fig.15. Effect of R/D ratio on the predicted penetration rate (CFD runs 2, 28-34): (a) penetration rate of the
T
whole pipe bends; (b) penetration rate of the elbows.
IP
Fig.16. The erosion location for the pipe bends as the Stokes number changes. Location A is the side wall of the
Fig.17. The relationship between β and the Stokes number.
SC R
elbow, location B is the outermost side of the elbow close to the elbow outlet.
AC
CE P
TE
D
MA
NU
Fig.18. Sample particle trajectories and velocity vectors for standard pipe bend.
32
ACCEPTED MANUSCRIPT
vp2 up1
T
up2
α2 (a)
TE
D
up1
MA
NU
vp2
SC R
α1
IP
vp1
CE P
vp1
up2
α2
α1
(b)
AC
Fig.1. Definition of the impact procedure of a particle: (a) straight wall; (b) curved wall.
33
MA
NU
SC R
IP
T
ACCEPTED MANUSCRIPT
AC
CE P
TE
D
Fig.2. Computational geometry used in the simulation.
34
D
MA
NU
SC R
IP
T
ACCEPTED MANUSCRIPT
AC
CE P
TE
Fig.3.Computational mesh used in the simulation.
35
MA
NU
SC R
IP
T
ACCEPTED MANUSCRIPT
AC
CE P
TE
D
Fig.4. Comparison of predicted velocity profiles with experimental data of Enayet et al.[41]
36
TE
D
MA
NU
SC R
IP
T
ACCEPTED MANUSCRIPT
Fig.5. Comparison of predicted and experimental penetration rate along elbow curvature angle for the five erosion
AC
CE P
models with different particle-wall rebound models.
37
MA
NU
SC R
IP
T
ACCEPTED MANUSCRIPT
AC
CE P
TE
D
Fig.6.Comparison of measured erosion from Bourgoyne [28] with different erosion models.
38
SC R
IP
T
ACCEPTED MANUSCRIPT
NU
Fig.7. Effect of inlet velocity on predicted penetration rate (CFD runs 1-5): (a) penetration rate of the whole
AC
CE P
TE
D
MA
pipe bends; (b) penetration rate of the elbows.
39
SC R
IP
T
ACCEPTED MANUSCRIPT
Fig.8. Effect of particle diameter on predicted penetration rate (CFD runs 2, 6-12): (a) penetration rate of the
AC
CE P
TE
D
MA
NU
whole pipe bends; (b) penetration rate of the elbows.
40
SC R
IP
T
ACCEPTED MANUSCRIPT
AC
CE P
TE
D
MA
NU
Fig.9. Erosion profiles and particle trajectories of pipe bend with different particle diameter:(a) 50μm； (b)150μm;(c)300μm.
41
SC R
IP
T
ACCEPTED MANUSCRIPT
Fig.10. Effect of pipe diameter on predicted penetration rate (CFD runs 2, 13-17): (a) penetration rate of the
AC
CE P
TE
D
MA
NU
whole pipe bends; (b) penetration rate of the elbows.
42
SC R
IP
T
ACCEPTED MANUSCRIPT
Fig.11. Effect of particles mass flow rate on predicted penetration rate (CFD runs 2, 18-22): (a) penetration rate
AC
CE P
TE
D
MA
NU
of the whole pipe bends; (b) penetration rate of the elbows.
43
SC R
IP
T
ACCEPTED MANUSCRIPT
NU
Fig.12. Effect of bend orientation on predicted penetration rate (CFD runs 2, 23-25): (a) penetration rate of the
AC
CE P
TE
D
MA
whole pipe bends; (b) penetration rate of the elbows.
44
SC R
IP
T
ACCEPTED MANUSCRIPT
NU
Fig.13. Effect of bending angle on predicted penetration rate (CFD runs 2, 26 and 27): (a) penetration rate of the
AC
CE P
TE
D
MA
whole pipe bends; (b) penetration rate of the elbows.
45
TE
D
MA
NU
SC R
IP
T
ACCEPTED MANUSCRIPT
AC
CE P
Fig.14. Flow field of the pipe bends in different bending angle.
46
SC R
IP
T
ACCEPTED MANUSCRIPT
NU
Fig.15. Effect of R/D ratio on the predicted penetration rate (CFD runs 2, 28-34): (a) penetration rate of the
AC
CE P
TE
D
MA
whole pipe bends; (b) penetration rate of the elbows.
47
SC R
IP
T
ACCEPTED MANUSCRIPT
NU
Fig.16. The erosion location for the pipe bends as the Stokes number changes. Location A is the side wall of
AC
CE P
TE
D
MA
the elbow, location B is the outermost side of the elbow close to the elbow outlet.
48
D
MA
NU
SC R
IP
T
ACCEPTED MANUSCRIPT
AC
CE P
TE
Fig.17. The relationship between β and the Stokes number.
49
TE
D
MA
NU
SC R
IP
T
ACCEPTED MANUSCRIPT
AC
CE P
Fig.18. Sample particle trajectories and velocity vectors for standard pipe bend.
50
ACCEPTED MANUSCRIPT Table captions
T
Table 1 Values of parameters in solid particle erosion models.
IP
Table 2 Experimental conditions from Zeng et al. [29].
SC R
Table 3 Experimental conditions from Bourgoyne[28].
AC
CE P
TE
D
MA
NU
Table 4 Conditions of the CFD runs.
51
ACCEPTED MANUSCRIPT Table 1 Values of parameters in solid particle erosion models.
b 22.7
α0 15
w 1
x 3.147
y 0.3609
DNV A1 9.370
A2 42.295
A3 110.864
A4 175.804
A5 170.137
A6 98.398
E/CRC R1 5.3983
R2 -10.1068
R3 10.9327
R4 -6.3283
R5 1.4234
Oka k0 65
k1 -0.12
k2 2.3(Hv)0.038
k3 0.19
n1 0.71(Hv)0.14
IP
SC R
NU
MA D TE CE P AC
52
z 2.532
T
Ahlert a -38.4
n2 2.4(Hv)-0.94
A7 31.211
A8 4.170
V’/(m/s) 104
d’/μm 326
ACCEPTED MANUSCRIPT Table 2 Experimental conditions from Zeng et al. [29].
Value
Fluid Velocity Particle diameter Particle density Particle mass flow rate Brinell hardness (BH)of pipe wall Pipe material density
Water 4m/s 450μm 2650kg/m3 0.235kg/s 160 7800kg/m3
AC
CE P
TE
D
MA
NU
SC R
IP
T
Name
53
ACCEPTED MANUSCRIPT Table 3 Experimental conditions from Bourgoyne[28].
Fluid
D(mm)
R/D
u0(m/s)
dp(μm)
ρp (kg/m3)
Mp (kg/s)
ρw (kg/m3)
BH
1 2 3
Water Water Water
52.5 52.5 52.5
1.5 3 3.25
9.45 11.49 14.63
350 350 350
2650 2650 2650
1.52 3.71 2.36
7800 7800 7800
120 140 140
AC
CE P
TE
D
MA
NU
SC R
IP
T
Case
54
ACCEPTED MANUSCRIPT
4
Water
5
Water
6
Water
7
Water
8
Water
9
Water
10
Water
11
Water
12
Water
13
Water
14
Water
15
Water
16
Water
17
Water
18
Water
19
Water
20
Water
21
Water
ρp (kg/m3)
Mp (kg/s)
ρw (kg/m3)
BH
90
40
1.5
5
200
2650
0.2
7800
140
90
40
1.5
10
200
T
Water
dp(μm)
2650
0.2
7800
140
90
40
1.5
15
200
IP
3
u0(m/s)
2650
0.2
7800
140
90
40
1.5
20
200
2650
0.2
7800
140
90
40
1.5
25
200
2650
0.2
7800
140
90
40
1.5
10
50
2650
0.2
7800
140
90
40
1.5
10
60
2650
0.2
7800
140
90
40
1.5
10
80
2650
0.2
7800
140
90
40
1.5
10
100
2650
0.2
7800
140
40
1.5
10
150
2650
0.2
7800
140
90
40
1.5
10
250
2650
0.2
7800
140
90
40
1.5
10
300
2650
0.2
7800
140
90
100
1.5
10
200
2650
0.2
7800
140
90
200
1.5
10
200
2650
0.2
7800
140
90
300
1.5
10
200
2650
0.2
7800
140
90
400
1.5
10
200
2650
0.2
7800
140
90
500
1.5
10
200
2650
0.2
7800
140
90
40
1.5
10
200
2650
0.05
7800
140
90
40
1.5
10
200
2650
0.1
7800
140
90
40
1.5
10
200
2650
0.15
7800
140
90
40
1.5
10
200
2650
0.25
7800
140
90
SC R
Water
R/D
NU
2
H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward
D(mm)
MA
Water
Bending angle(°)
D
1
Orientation
CE P
Fluid
AC
CFD run
TE
Table 4 Conditions of the CFD runs.
55
27
Water
28
Water
29
Water
30
Water
31
Water
32
Water
33
Water
34
Water
2650
0.3
7800
140
90
40
1.5
10
200
2650
0.2
7800
140
90 90
40 40
1.5 1.5
10 10
200 200
2650 2650
0.2 0.2
7800 7800
140 140
45
40
1.5
10
200
T
Water
200
2650
0.2
7800
140
180
40
1.5
10
200
IP
26
10
2650
0.2
7800
140
90
40
2
10
200
2650
0.2
7800
140
90
40
2.5
10
200
2650
0.2
7800
140
90
40
3
10
200
2650
0.2
7800
140
90
40
4
10
200
2650
0.2
7800
140
90
40
5
10
200
2650
0.2
7800
140
90
6
10
200
2650
0.2
7800
140
40
8
10
200
2650
0.2
7800
140
40
90
SC R
Water Water
1.5
NU
24 25
40
MA
Water
90
TE
23
H-V downward V-H downward V-H upward H-V upward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward H-V downward
CE P
Water
AC
22
D
ACCEPTED MANUSCRIPT
56
ACCEPTED MANUSCRIPT
SC R
IP
T
Graphical Abstract
AC
CE P
TE
D
MA
NU
Three types of collisions occur due to the change of Stokes numbers in liquid-solid flow, namely, secondary flow driven collision, direct collision and sliding collision.
57
ACCEPTED MANUSCRIPT Highlights
The two-way coupled Eulerian-Lagrangian approach is employed to solve the liquid-solid flow
The relationship between Stokes number and the dynamic movement of the maximum
IP
T
in the pipe bend.
Three collision mechanisms are proposed to explain how the changes of Stokes numbers
CE P
TE
D
MA
NU
influence the erosion location.
AC
SC R
erosion location is presented
58