Investigation of erosion behaviors of sulfur-particle-laden gas flow in an elbow via a CFD-DEM coupling method

Accepted Manuscript Investigation of erosion behaviors of sulfur-particle-laden gas flow in an elbow via a CFD-DEM coupling method

Dezhi Zeng, Enbo Zhang, Yanyan Ding, Yonggang Yi, Qibiao Xian, Guangju Yao, Hongjun Zhu, Taihe Shi PII: DOI: Reference:

S0032-5910(18)30069-X https://doi.org/10.1016/j.powtec.2018.01.056 PTEC 13142

To appear in:

Powder Technology

Received date: Revised date: Accepted date:

10 May 2017 14 January 2018 22 January 2018

Please cite this article as: Dezhi Zeng, Enbo Zhang, Yanyan Ding, Yonggang Yi, Qibiao Xian, Guangju Yao, Hongjun Zhu, Taihe Shi , Investigation of erosion behaviors of sulfur-particle-laden gas flow in an elbow via a CFD-DEM coupling method. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Ptec(2017), https://doi.org/10.1016/j.powtec.2018.01.056

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ACCEPTED MANUSCRIPT Investigation of erosion behaviors of sulfur-particle-laden gas flow in an elbow via a CFD-DEM coupling method

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Dezhi Zenga*, Enbo Zhanga, Yanyan Dingb, Yonggang Yib, Qibiao Xianc, Guangju Yaoc, Hongjun Zhua, Taihe Shia a State Key Laboratory of Oil & Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, China b Engineering Technology Research Institute of Xinjiang Oilfield Company, Karamay 834000, China c Southwest Oilfield Company, Sinopec, Chengdu, 610051, China * Corresponding author. E-mail address:[email protected] Fax:028-83032901 Abstract: In the production and gathering process of natural gas with the high sulfur content, due to the variations in temperature, pressure, and other factors, sulfur dissolved in the gas may be precipitated as a solid particle in the gathering pipeline. Sulfur particles carried by high-speed flow impact elbow of pipelines, thus causing equipment malfunctions and even failures. In this study, a CFD-DEM-based erosion prediction model for gas-particle two-phase flow was proposed based on the consideration of the gas-particle, particle-particle and particle-wall interactions. The effects of secondary flow, vortices and particle trajectories on rare erosion scars were investigated. In addition, four kinds of polyhedral particles were modelled based on DEM framework to simulate erosion behaviors based on the consideration of particle shapes. The results indicate the V-shaped erosion scar is caused by the secondary collision. Two adjacent obvious erosion scars on the upper part of V-shaped scar are caused by direct collisions and sliding collisions. From the inlet to the outlet of the elbow, the turbulence intensity near the two sides of the wall increases and the secondary flows and vortices appear. Particle trajectories are affected by complex flow fluid, which causes rare erosion scars on the side walls of the downstream straight pipe near the elbow outlet. With the increase in the particle sphericity, the erosion rate decreases firstly and then increases. When the sphericity is less than 0.77, the erosion rate is mainly affected by the impact velocity and impact angle; when the sphericity is larger than 0.77, the influence of impact concentration on the erosion rate is more obvious. Keywords: Sulfur particle; Elbow erosion; Gas-solid flow; CFD-DEM; Particle shape; Numerical simulation

1 Introduction Erosion corrosion, also named wear corrosion, refers the phenomenon of serious damage caused by solid particles carried by high-speed flow. In the process of production and gathering of natural gas with the high sulfur content, hydrogen sulfide is converted into sulfur atoms, which are condensed to form sulfur particles due to the changes in temperature, pressure, and other factors. Hyne [1] suggested that there

ACCEPTED MANUSCRIPT were two kinds of solid sulfur particles in gas reservoirs: Firstly, in high-temperature, high-pressure and high-sulfur gas reservoir, the slow degradation of hydrogen sulfide or the reduction reaction between ferrous sulfate and hydrogen sulfide produces elemental sulfur: FeS2 H2S   H 2 +S 

(1)

H2S+FeS2  FeS+S  +H2S（H2Sx）

(2)

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149℃

Secondly, under the conditions of high temperature and high pressure, carbon

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dioxide reacts with hydrogen sulfide to produce elemental sulfur:

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200℃, 137.9MPa CO2 +6H2S  S  10Days

(3)

Chesnoy and Pack [2] suggested the two possible reactions of solid sulfur in the

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gathering system:

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2H2S(g) +O2(g)  2S(s) +H2O(g) C6H6(g) +4H2S(g)  2C6H14(g) +4S(s)

(4) (5)

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Partial elemental sulfur is precipitated and deposited in the formation and well

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bore, whereas partial elemental sulfur is carried above the ground in the gaseous and solid states by natural gas. The solid sulfur particles coming out of the well bore are be captured by the separator, but the sulfur element in the gaseous state is precipitated

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in the gathering and transportation system. Therefore, sulfur deposition is a problem which cannot be ignored in the exploitation of high sour gas fields. Deposited

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particles block the production layer and reduce the productivity of gas wells and the collisions between particles and the inner wall of the pipeline caused erosion corrosion. Particularly, in some pipeline fittings such as elbow, erosion could be more serious due to the effects of pressure drop, centrifugal action and dramatic change in the flow field. The thickness loss due to erosion on elbows is about fifty times more than that on straight pipelines [3]. Erosion severity is determined by various factors such as flow velocity, multiphase flow regimes, particle characteristics, and fluid properties. Pipeline erosion under different conditions had been experimentally investigated. Kesana [4]

ACCEPTED MANUSCRIPT experimentally investigated the influences of the parameters of horizontal slug flow on erosion rates. Fan [5] studied the effect of liquid flow rate on the erosion rate of a pipe with a large diameter with Electrical Resistance (ER) probes. Vieira [6] studied the effect of particle speed on the erosion rate with a particle image velocimetry (PIV) technique. However, the limitations of experimental conditions, measurement errors,

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and the operating errors may affect the experimental results. A simulation technology has become an effective research method due to its main advantages of low cost and

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simple control.

In recent years, Computational Fluid Dynamics (CFD) technology has been

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widely used in many industrial fields and the erosion research methods based on CFD are widely used. In order to predict the erosion accurately, two effective particle-scale

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models had been developed to investigate particle movements in an elbow: CFD-DPM (computational fluid dynamics-discrete phase model) [7] and CFD-DEM

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(computational fluid dynamics-discrete element method) [8]. CFD-DPM method is widely used in simulations of liquid-solid and gas-solid

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flows. However, CFD-DPM is based on the assumption that the interaction between

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particles can be neglected, so it is suitable for dilute flow [7]. Wang [9] developed an erosion model for long-radius elbow based on CFD-DPM. Njobuenwu et al. [10] took particle–wall interaction into account and predicted the erosion of dilute flow in 90°

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square cross-section elbow with a Lagrangian particle tracking approach. Chen et al. [11] investigated elbow erosion behaviors in dilute gas-solid flow with CFD-DPM

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and developed an erosion model for multiphase flow without considering the particle-particle interaction [12]. Mazumder [13] studied the effects of gas/liquid velocity on erosion locations in a U-shaped elbow with CFD-DPM. Although the particle-particle interaction was neglected, these researchers introduced the particle-wall rebound model and obtained the consistent results with experimental data. In the above studies, CFD-DPM method was used to explore dilute flow, but dense flow conditions are frequently encountered in the production process. CFD-DPM method is not suitable for dense flow. With the development of particle simulation technique in recent years, the

ACCEPTED MANUSCRIPT discrete element method (DEM), which can simulate single particle motion, has become one of the mainstream discrete modeling techniques. Thus, the interaction between particles can be considered in CFD-DEM method for erosion prediction [14]. Varga et al. [15] simulated multiphase flow in feed pipes with CFD-DEM method and obtained the consistent numerical results with experimental data. Chu and Yu [16, 17] combined CFD code with DEM software to simulate liquid-solid flow in the 3D fluid

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system. Chu [18] combined CFD-DEM with Finnie’s erosion model to calculate

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erosion rate of dense medium cyclones in multiphase flow. The above studies indicated that the CFD-DEM method could be effectively used to predict the erosion

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behaviors of dense particle flow.

In previous studies on elbow erosion, sand or other hard particles were treated as

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solid phase [19-24]. However, compared with sand, sulfur particles have different characteristics such as smaller density and lower hardness and are susceptible to the

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flow field. The erosion effects of sulfur particles on elbows or trajectories of sulfur particles in elbows are rarely reported. At present, some scholars have studied the

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erosion behaviors of the particles similar to sulfur particles. Mansouri et al. [25]

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investigated erosion behaviors of quartz particles in sharp elbows using numerical and experimental method. Rawat et al. [26] studied erosion wear caused by high-concentration fly ash slurries and indicated that the incident angle of 45° allowed

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the maximum erosion rate. Jin et al. [27] adopted the immersed boundary method to study the erosions caused by fly ash particles on a staggered tube bank.

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In the above studies performed with CFD-DPM and CFD-DEM methods, spheres were used to represent particles in order to simplify the physical process of particle-particle and particle-wall interactions. However, in the gathering and transportation system, the erosion of the elbow is also affected by the particle shape, which significantly affects the impact angle due to geometric boundary. Some researchers investigated the model of irregular particles [28, 29]. Ben-Ami [30] suggested a semi-empirical erosion model to predict the erosion rate of a single particle and particle diameter was a function of particle shape in this model. The effects of particle shape, impact angle and angular velocity on the shear energy were

ACCEPTED MANUSCRIPT analyzed through numerical simulation [31]. It is necessary to precisely explore the flow characteristics of conveying medium in elbows and the influence of particle shape and trajectory on erosion behaviors for the maintenance and inspection of equipment in the production field of high sour gas field. In this work, based on CFD software ANSYS-FLUENT 17.0 and DEM code EDEM 2.7, the CFD–DEM coupling approach was employed to investigate the

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erosion behaviors of particle-laden gas in a 90o elbow. In order to verify the model

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and approach adopted in this study, the results of a verification case were compared with experimental results [32]. The gas–particle, particle–particle and particle–wall

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interactions were considered in the study. In addition, the causes for rare erosion scars were analyzed in the following aspects: particle trajectories, secondary collision, and

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secondary flow vortices. Furthermore, four kinds of polyhedral particles (tetrahedron,

of particle shapes on erosion rate.

2 Numerical model

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hexahedron, octahedron, and dodecahedron) were modelled to investigate the effects

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The numerical simulation involves three models: flow field model, particle

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motion model, and erosion model. 2.1 Flow field modeling

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In order to save calculation time, a one-way coupling method is often employed to perform CFD-DEM simulations. But in the one-way coupling method, the force of

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particles applied on the fluid is ignored. The force of particles applied on the fluid is considered in the two-way coupling method. The two-way coupling method was used in this research. The motion of the flow fluid can be obtained with the local mean variables according to the continuity and momentum conservation equations. The gas velocity is not high and the pressure distribution in the elbow is basically uniform, so the flow is defined as incompressible flow. The governing equations of the fluid are given as follows: Continuity equation

ACCEPTED MANUSCRIPT   ( f  f )  ( f  f u j )  0 t x j

(6)

Momentum conservation equation

  p ( f  f ui )  ( f  f uiu j )   t x j xi

f

(7)

represents the fluid density and is a constant because the fluid is assumed

u

represents the fluid velocity;

p

represents the pressure of

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to be incompressible;

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where

 ui u j    (  )    f  f g  Fs  f eff  x  x  j i  

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  x j

the fluid; eff represents the effective viscosity;

x

represents the coordinates;

g

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is the acceleration due to gravity; Fs is the interaction term due to the drag force

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between the particles and the fluid;  f is the porosity near the particle and can be calculated as:

n

where

(8)

i 1

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 f  1  Vp ,i / Vcell

V p ,i represents the volume of particle i in the selected CFD cell; n represents

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the amount of particles inside the cell;Vcell represents the volume of the cell. 2.2 Discrete particle motion modeling

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The flow as a continuum phase is simulated by the local averaged Navier–Stokes equations. The motion of particles is defined as a discrete phase and described with the Newton's laws of motion. Particle may impact other particles and wall in the movement. Particles in flow fluid are affected by gravity, drag force, and so on. The translational and rotational movements of a particle can be calculated with Newton's Kinetic Equation:

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dv  mg   Fc  Fdrag  Fb  Fm  Fsl  Fpg  Fvm dt

(9)

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I where

dω   Tc  Tf dt

Fc , Fdrag , Fb , Fm , Fsl , Fpg ,

Fvm

and

(10)

respectively represent contact force,

fluid drag force, basset force, magnus force, Saffman’s lift force, pressure gradient force and virtual mass force;

m

and

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are respectively the mass and moment of

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inertia of the particle; dv dt is the translational acceleration of the particle; dω dt

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is the angular acceleration of the particle; Tc and Tf are respectively the contact

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torque and the torque caused by the fluid.

The torque from fluid on a rotating particle can be calculated as follows [21]:

f dp (

)5 CR ω ω

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Tf =

2

2

(11)

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where the coefficient of the rotation torque can be obtained as follows [21]:

(12)

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64  ,Rer  32  Rer  CR    12.9  128.4 ,32  Re  1000 r  Re1/2 Rer r

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Where Rer is the Reynolds number of the particle and defined as:

 f d p2 ω Rer  f

(13)

ω  0.5  v f  ω p

(14)

where ω p is particle rotation velocity. The basset force, Magnus force, pressure gradient force and virtual mass force are very small, and thus forces are neglected in this work. This will be validated by additional explanations in the following sections. So the Eq.(9) can be converted to Eq.(15):

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dv  mg   Fc  Fdrag  Fsl dt

(15)

ACCEPTED MANUSCRIPT As shown in Fig. 1(a), in a soft-sphere model, particle i affected by fluid contacts with particle j at point E. In the subsequent motion, two particles are deformed. δ n and δt

are respectively Normal displacement and tangential

displacement. Fc ,n and Fc ,t are respectively the normal contact force and the tangential contact force. The constitutive model of the interaction between two

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particles is shown in Fig. 1(b).

The spring can simulate the deformation damp and the damper can simulate the

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damping effect. If the diameter of a particle is infinite, this model can reflect the

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collision between particle and wall.

The contact force Fc in Eq. (15) can be calculated as follows:

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Fc  Fc,n  Fc ,t

(16)

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where Fc ,n and Fc ,t are respectively normal component and tangential component. The linear spring-dashpot model suggested by Cundall and Strack [33] was used to

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calculate the contact forces Fc ,n and Fc ,t between two particles:

Fc,t  kt δt  t vt

(18)

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(17)

are respectively the normal displacement and tangential

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where δ n and δt

Fc,n  knδn  n vn

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displacement between the particles; v n and v t are respectively normal and tangential relative velocity between the particles; kn and kt are respectively the normal stiffness and tangential stiffness of the spring;

 n and t are respectively the normal and

tangential damping coefficients. Ting et al. [34] proposed equations to calculate damping coefficients with the coefficient of restitution. For the tangential contact force, the Coulomb friction model [35] is applicable when the two components of the contact force meet the following conditions:

Fc,t  f s Fc ,n

(19)

ACCEPTED MANUSCRIPT The tangential contact force can be calculated as follows:

Fc,t   f s Fc ,n δt δt

(20)

where f s is the sliding friction coefficient. The contact torque Tc is equal to the torque Tt created by the tangential contact force:

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Tc  Tt

(21)

where n is the normal unit vector;

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Tt  rn  Fc,t

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The tangential contact torque Tt is defined as follows:

(22)

r is the radius of the spherical particle.

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Among all the forces applied on particles by the flow, the drag force shows the most significant influence on the particles. Wen-Yu and Ergun Drag Force Model [36]

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can be used to describe effectively the interaction between fluid and particles, and Haider and Levenspiel [37] Drag Law was adopted:

where

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 150 (1   )  18  2 ,   0.8 Fdrag    Re C (Re) 3.65 ,   0.8  24 d b Re 24 Cd 0 = 1  b1 Rebp2  3 p Re p b4  Re p





(23)

(24)

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b1  exp  2.3288  6.4581  2.4486 2  b1  0.0964  0.5565

b1  exp  4.905  13.8944  18.4222 2  10.2599 3 

(25)

b1  exp 1.4681  12.2584  20.7322 2  15.8855 3  Cd 0 ,  ,  and Re p , are respectively the drag coefficient, the inter-phase momentum exchange coefficient [36] and Reynolds number of the particle and the sphericity of a particle [38]:

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Re p 

 f f dp v f  vp

(26)

f  

Ss Sp

(27)

Where S s is the superficial area of a sphere (m2); S p is the superficial area of a

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non-spherical particle (m2). The virtual mass force is

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The pressure gradient force is

dv f Fpg V pp dt  ~ dv dv dv m p p  pVp p  p p dt dt dt

f

(29)

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Saffman’s lift force [39] is

(28)

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d (v f  v p ) 1  Vp f Fvm 1  f d (v f  v p ) dt 2  dv dv 2 p dv p mp p  pVp p dt dt

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  Fsl  1.615d p2  f f  ωf 

1/2

  Csl  v f  v p   ω f   

(30)

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where ω f is the fluid rotation velocity, which can be defined as follows:

ωf   v f

(31)

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Csl is the lift force coefficient and can be calculated according to Mei [40]:

1  0.3314 1/2  eRe p /10  0.3314 1/2 ,Re p  40  Csl   1/2 0.0524  Re ,Re p  40    p 



dp ω f 2 v f  vp

where Rer is the Reynolds number of the particle and defined as:

(32)

(33)

ACCEPTED MANUSCRIPT  f d p2 ω Rer  f

(34)

ω  0.5  v f  ω p

(35)

where ω p is particle rotation velocity. 2.3 Erosion modeling

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The erosion mechanism is complex and involves various factors. A general erosion model considering all the factors is not available. At present, most of classical

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erosion models are established based on the empirical formula, and those models

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include different unit of erosion rate. Hence, it is necessary to select an erosion model considering more factors as possible and exchange the unit of erosion rate.

e1r M W Kf ( ) MvP W = mAface PW Aface

(36)

mvP2 e  Kf ( ) PW

(37)

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1 Rerosion 

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The Finnie erosion model [42] is defined as follows:

1 r

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2  sin 2  4sin  (  14.04 ) f ( )=  2 (  14.04 )  cos  / 4

1

(38)

where Rerosion is the erosion rate given in units of the mass of material loss per

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unit area per unit time; K is a dimensional constant depending on the particle and 1

wall properties; er is the erosion rate of wall in Finnie model;

M is the mass flow

W is the density of wall material; m is particle mass; v p is the

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rate of particle;

impact velocity of particles; PW is the flow stress of wall; Aface is the area of the cell face; f ( ) is the function of impact angle. The E/CRC model [43] is defined as follows: 2 erosion

R

er2 M  Aface

er2  Kf ( ) FS  vP 

(39) n

(40)

ACCEPTED MANUSCRIPT 2  (  10 ) a  b f ( )=  2 2  C1 cos  sin(C2 )  C3 sin   C4 (  10 )

2

(41)

2

where Rerosion is the general erosion rate; er is the erosion rate of wall in E/CRC model which is defined as the mass removed from the surface per mass of

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particles impinging on the surface; FS is the particle shape coefficient and values of 0.2, 0.53 and 1 for rounded, semi-rounded and sharp particle, respectively; a, b, C1, C2,

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C3, C4 and n are constant depending on the particle and wall properties. The Oka erosion model [44, 45] is defined as follows:

e  K (aHv)

k1b

(42) k

k2 3  vP   d p  f ( )      v   d  

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3 r

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R

er3 M W  Aface

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3 erosion

f ( )=  sin   1 1  Hv(1-sin ) 2 n

n

3

(43) (45)

3

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where Rerosion is the general erosion rate; er is the erosion rate of wall in Oka

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model; Hv is the Vickers hardness of the wall surface; d  is the reference diameter; v is the reference velocity; a, b, k1, k2, k3, n1 and n2 are constant

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depending on the particle and wall properties. In this study, the sphericity of non-spherical particle was taken into account to

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modify the erosion model. The modified erosion model is defined as follows:

3

    Rerosion Rerosion f ( )

(46)

3

where Rerosion is the general erosion rate; er is the erosion rate of wall in Oka model; Hv is the Vickers hardness of the wall surface; d  is the reference diameter; v is the reference velocity; a, b, k1, k2, k3, n1 and n2 are constant depending on the particle and wall properties, the dimensionless constant  accounts for erosion models, and takes values of 1, 2 and 3 for modified Finnied erosion model, modified E/CRC model, and modified Oka erosion model.

ACCEPTED MANUSCRIPT 3 Model implementation The CFD-DEM coupling method considering the gas-particle, particle-particle and particle-wall interactions was employed for simulation in the study. The SIMPLEC method and transient-in-time solver were adopted to solve the governing equations for continuous phase flow in CFD software ANSYS Fluent 17.0. The

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turbulence model was then tested and the standard wall functions for near-wall zone treatment were used. The DEM code EDEM 2.7 coupled with CFD through user

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defined functions (UDFs) was adopted to describe the motion of solid phase. The time

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step of EDEM was set to be smaller than the Fluent’s time step according to the suggestions of Ting and Corkum [46].

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3.1 Geometric model and mesh

In the study, an elbow of common pipelines was as an example. The inner

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diameter of the elbow is D=40 mm; the elbow radius of curvature is R/D=1.5. In order to realize a fully developed flow and a sufficient dispersion of the solid phase, the 10D-length (400 mm) horizontal and vertical pipelines were added at the inlet and

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outlet of the elbow. The wall material is carbon steel with a Brinell hardness of 120.

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The schematic diagram of computational domain is shown in Fig. 2. The mesh-generator ICEM CFD was used to generate the geometric model and meshes. The hexahedral structured mesh was adopted in the simulation to ensure the

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higher stability and accuracy. The Reynolds number of simulations was high, so the y+

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was set as 30. The progressive mesh was adopted to mesh the boundary with ten-layer grids in the radial direction. The height of the first cell was 0.05 mm and the growth factor was 1.2. Moreover, a finer mesh was employed in the elbow section to obtain information of flow accurately. To ensure that the number of grids would not affect the results, the sensitivity tests of various mesh numbers were performed through the standard case. The parameters of the standard case are shown in Table 1. The erosion rate of extrados was employed to test the sensitivity of different meshes (Fig. 3). With the increase in the number of meshes, the erosion rate was slightly changed. The erosion rate did not show any significant variation when the number of the meshes

ACCEPTED MANUSCRIPT was more than 708564. Moreover, the number of the meshes was set as 708564 for further simulations in the model. 3.2 Comparison among various turbulent models The Reynolds number in the simulation case was high. Various turbulence models including standard k-epsilon model, RNG k-epsilon model and realizable

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k-epsilon model were compared with the standard case. The ratios of local velocity (V) to mean velocity (V0) along the center line under different models for the standard

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case were obtained (Fig. 4). The ratio shows a slight difference from inlet to outlet of the elbow (400 mm~500 mm) among three models. This difference may be attributed

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to the change in the turbulent intensity of flow. With the increase in the elbow angle, the turbulence intensity gradually increases. When the fluid flows out of the elbow

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(>500 mm), the intensity decreases and the velocity difference is small. However, the ratio shows a slight variation among the three turbulence models. Hence, the

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influences of various turbulence models on the simulation results may be neglected. The standard k-epsilon model was adopted in the subsequent investigation.

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3.3 Interaction between particle and fluid

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The forces applied on the particles in the flow mainly include drag force, Saffman’s lift force, virtual mass force, and pressure gradient force. In order to investigate the influences of the latter three forces on erosion rate, a comparative test,

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was performed with the essential parameters of the standard case (Fig. 5). The influences of Saffman’s lift force the distribution of erosion rate is obvious,

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but the effects of virtual mass force and pressure gradient force are insignificant. Saffman’s lift force is related to the changing velocity gradient in the flow field. Moreover, the change in velocity gradient near the wall is significant and its influence on particles also becomes significant. If the flow field is not dense, the pressure gradient in the flow field is small. Due to the small scale of particles, the variation of pressure within the distance of particle diameter is slight. Therefore, the pressure gradient force is neglected. The virtual mass force is proportional to the continuous phase density, hence, it is most significant when the dispersed phase density is less than the continuous phase

ACCEPTED MANUSCRIPT density. In this research, the dispersed phase (solid particle) density is larger than continuous phase (gas) density. So the influence of virtual mass force was neglected in this study. The basset force is the instantaneous resistance when the particles are accelerated in a viscous fluid. The magnus force is produced by the rotation of particles. When a particle has a large diameter with an irregular shape, the force is

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obvious. Hence, the drag force and the Saffman’s lift force were considered in simulations. Of course, both flow and particles were affected by gravity.

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3.4 Validation of the model

In order to select the appropriate erosion model and verify that model in this

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study, three modified erosion models, Finnied erosion model, E/CRC model and modified Oka erosion model, were adopted to calculate the verification case, and the

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more appropriate model was compared with experimental results [32]. The conditions of the verification case are basically consistent with the experimental conditions

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(Table 2).

A great number of particles require a lot of computational resource for simulation.

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Thus, only the last set of experimental conditions were compared with simulation

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results.

The simulation conditions were basically consistent with experimental conditions. The erosion data along the centerline of the extrados of elbow is obtained in the same

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way in the experiment. For unification, the dimensionless parameter called relative erosion rate was presented in this work, which is defined as the ratio of erosion rate in

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each computational location to the maximum one of erosion results. As shown in Fig.6, the erosion results of three models are similar with each other. Additionally, Fig.7 shows the standard deviation (SD) of each model result and the average of three models results. It performs that the standard deviation of Finnie erosion model is the minimum. Hence, the modified Finnie erosion model is used to investigate the erosion caused by non-spherical particles. Fig. 8 shows the comparison results, using the modified Finnie erosion model, between the erosion rate and the experimental data from the inlet (0°) to the outlet (90°) of the elbow [32]. The maximum erosion rate obtained in the experiment, 4.38

ACCEPTED MANUSCRIPT ×10-5 kg/(m2·s), was obtained at the position (48°) of the elbow. The maximum erosion rate obtained through numerical results, 4.29×10-5 kg/(m2·s), was obtained at the position (47°) of the elbow. The simulation results were consistent with experiment results. Therefore, the model and method used in this study can effectively predict elbow erosion.

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4 Results and discussion 4.1 Simulation results of the standard case

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The gas flow affects particle motion significantly. Fig. 9(a) and Fig. 9(b) respectively illustrate the velocity and pressure profiles of the fluid inside in the X–Z

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plane of the elbow. In the field of velocity, methane flows into the horizontal pipeline in the velocity of 20 m/s along the X axis. The gas near the extrados of the elbow is

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decelerated, whereas the gas near the intrados is accelerated. Besides, a low-velocity region appears near the outlet of the elbow. This difference may be interpreted as

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follows. When gas flows into the elbow, the pressure distribution is changed under the effect of centrifugal force. The maximum pressure can be obtained near the extrados,

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whereas the minimum pressure can be observed near the intrados.

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Fig. 10 illustrates the position and motion of particles at different time points (t=0.005 s, t=0.01 s, t=0.015 s, t=0.02 s, t=0.025 s, and t=0.03 s) after the sulfur particles enter the horizontal pipeline at t=0. Particle colors represent instantaneous

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particle velocities. When t=0, the flow field is not fully developed at this time and the velocity varies among different particles. When t=0.005 s, the flow field is fully

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developed and the velocity of all particles is basically the same as that of the gas flow rate. Therefore, it is reasonable that the 10D-length (400 mm) horizontal and vertical pipelines are added at the inlet and outlet of elbow. When t=0.015 s, the particles have collided the wall, the particle velocity decreases and particles bounce back in other directions. In order to investigate the movement of particles in the flow field and the cause of the erosion scars, the relationship between the erosion scars and the trajectories of particles was analyzed in detail. Fig. 11 shows the erosion distributions of the elbow under the conditions of the

ACCEPTED MANUSCRIPT standard case. The maximum erosion rate, 2.18×10-7 kg/(m2·s), occurs at the position (51°) of the elbow. An obvious V-shaped erosion scar occurs below the area with the maximum erosion rate. The result is consistent with the report by Pereira et al. [47]. They considered that the V-shaped scar was attributed to the sliding collision of particles with the lower velocity [47]. However, in this study, the cause for the

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V-shaped scar is different. Fig.12 illustrates the trajectories of sulfur particles. The first collision occurs

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between high-speed particles and the inner wall of the elbow (red dashed area in Fig. 12). After the first collision, particle velocity reduces obviously and partial energy of

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particles is transferred and lost. The particle trajectories are drastically changed at the collision region (red dashed area in Fig. 12), and the most of particles are rebounded

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in two major directions (red arrows) after the particles impact inner wall. The secondary collisions occur between rebounded particles and inner wall and particle

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movement directions are aligned with the V-shaped erosion scar. Therefore, the V-shaped erosion scar is attributed to the secondary collision between the rebound

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particles and wall. The cause is different from the result reported by Pereira et al. [47].

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Maximum erosion rate occurs in Area A (red dashed graph) and there are two obvious red erosion scars in Area A. The distance between these two scars is short due to the effects of sliding collision and direct collision.

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As shown in Fig. 13(a), Particle E and Particle F enter the elbow from Point c and Point f, respectively. Particle E moves along the negative direction of Z axis for

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the distance of Δx under the effect of gravity and impacts the wall at Point d. Particle E does not bound off the wall into the bulk flow, but continues to move along the downstream because the impact angle α is small. Moreover, Particle E slides and impact inner wall at Point e. Particle F enters the elbow, impacts the wall directly at Point g, and then bound into the bulk flow of downstream. The distance between Points e and g is small and an obvious erosion scar occurs near the two points. Therefore, two obvious red erosion scars in Area A are attributed to effects of sliding collisions and direct collisions. One scar is near to Point d and the other one is near to Points e and g.

ACCEPTED MANUSCRIPT Fig. 11 illustrates rare erosion scars (red dashed ovals) on the side walls of the downstream straight pipe near the elbow outlet. However, in most previous studies, erosion scars are mainly concentrated in the extrados and outlet of intrados [48, 49]. The scars on the side wall is produced by the secondary flow vortices. Most of the particles enter the vertical straight pipeline after impacting the inner wall of the elbow,

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and only a few particles collide with the side walls of elbow outlet because the particle-free zone (red polygon) is near to the intrados of the elbow, as shown in Fig.

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13(b). In this zone, most particles will not collide with inner wall. The particle-free zone is produced by the effect of shadow, as indicated by Sommerfeld and Huber

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[50].

In order to determine whether erosion scar on the side wall is caused by the

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secondary flow vortices, the turbulent intensity, streamlines, and velocity vectors were acquired in a cross-section normal to the elbow axis at 0°, 30°, 60°, and 90° and 1D,

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downstream (D is inner diameter of the elbow) of the elbow outlet, as shown in Fig. 14 to Fig. 16. Fig. 14 shows the contours of turbulent intensity in five cross-sections.

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With the increase in the elbow angle, the variation of turbulent intensity is more

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obvious. At the angle of 60°, the turbulent intensity is changed significantly near the side wall of the elbow. When the flow reaches the outlet of the elbow (90°), the region with maximum turbulent intensity is located on the side walls of the elbow and

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becomes larger. The fluid continues to flow to the downstream and the secondary flow vortices move from the side walls to the center of the flow field. When the flow

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reaches the 1D downstream of elbow outlet, the region with the maximum turbulent intensity is located in the center of bulk flow, as shown in Fig. 14(e). Even though the geometric model of the elbow is simple, the flow regime in the elbow is complex and the turbulent intensity of fluid is changed dramatically. In order to analyze the generation and development of vortices, the streamlines in 5 cross-sections were acquired. The streamlines along vortical corelines can be clearly observed (Fig. 15). Firstly, at the plane (a), when the flow enters the elbow, there is no vortex. Secondly, at the plane (b), the two opposite vortices appear near the side walls in a symmetric way while the spiraling vortex motion starts on the wall. Thirdly, at the

ACCEPTED MANUSCRIPT planes (c) and (d), these two vortices become larger and gradually approach the symmetry axis. Finally, in the 1D downstream region, vortices continue to develop near the region of bulk flow, as shown in Fig. 15(e). The velocity vectors at various cross-sections are shown in Fig. 16. The paths of the secondary flow can be obviously observed. Initially at the plane (a), the directions

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of velocity vectors are coincident. A couple of opposite vortices are formed at 60° cross-sections and partial flow is separated from bulk flow to form secondary flow. As

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shown in Fig. 16(d) and Fig. 16(e), when the fluid moves to the outlet of the elbow and flows to downstream vertical pipe, the two opposite vortices are more obvious.

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Cross-sections normal to the elbow axis were acquired at 0°, 30°, 60°, 90°, and 1D downstream of the elbow outlet and the contours of turbulent intensity,

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streamlines and velocity vectors in these sections were analyzed, respectively. Therefore, it is reasonable that the erosion scar on the side wall is caused by the

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secondary flow vortices.

4.2 Effect of particle shape on erosion

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To investigate the relationship between particle shape and elbow erosion, four

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common polyhedrons, tetrahedron, hexahedron, octahedron and dodecahedron, were modeled in the study (Fig. 17). The properties of polyhedrons are listed in Table 3. The modeled macro particles with different geometrical features were overlapped by

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micro spheres with DEM framework. Sphericity describing the level that a particle shape is close to the pattern of a

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sphere is employed to represent the geometry of non-spherical particle. The sphericity is defined as the ratio of the superficial area of a sphere to the superficial area of the non-spherical particle with the same volume as the sphere, which is defined as Eq.(27). The schematic motions of particles with different shapes and the forces are shown in Fig. 18. On the basis of previous researches, the influences of particles on erosion are not only affected by the materials of particles and the wall, but also affected by impact velocity, impact angle and impact concentration. Impact concentration is defined as the degree of collisions on a per unit area. It increases with

ACCEPTED MANUSCRIPT the particle sphericity because particles with less faces have the longer lengths between two adjacent impact points, as shown in Fig. 18. Thus, collision concentration increases with the increase of particle sphericity and the increasing rate of collision concentration becomes slow when the sphericity is relatively large. In addition, particle velocity is the same to the velocity of gas before the continuous

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collisions appear on the wall. After the first impact between the particle and the wall, some particles may roll and bounce along the outer wall of the elbow. The particles

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with the long bounce distance are subjected to more powerful drag force, so particles have a larger impact velocity. Hence, with the increase in particle sphericity, the

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impact velocity decreases. Furthermore, when the particles bounce along the wall, the energy decreases rapidly and the bounce height and bounce distance decrease.

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Therefore, with the increase in the particle sphericity, particle collision velocity decreases.

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To compare the influences of different particle shapes on elbow erosion, the erosion behaviors of 4 kinds of polyhedral particles (  =0.684,



=0.843,

=0.912) were simulated through the standard case. The variation of erosion

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

=0.796,

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and



rates for four regular particles on the elbow are shown in Fig. 19. It is assumed that different particles have the same density and that the difference in the micro-particle

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density is neglected. The erosion rate is small between 0 (inlet) to 30° of extrados and its minimum value is 0. When the degree is larger than 30°, the erosion rate increases

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firstly and then decreases, and the regular increasing trend is obvious. The erosion results on the middle part of extrados are complicated. The maximum values of different particles except tetrahedron occur at approximately 51°. The secondary peak

values of different particles occur near 57° except that secondary peak value of tetrahedron occurs near 60°. The phenomenon of two peak values is consistent with the two obvious red erosion scars in Area A. The phenomenon may be ascribed to secondary collision and direct collision. The second peak value of tetrahedron slightly lags behind because the length between two adjacent impact points of tetrahedron is longer.

ACCEPTED MANUSCRIPT As shown in Fig. 19, the erosion rate of tetrahedron is lower than the results of the other particles. The erosion rate of dodecahedron is higher than that of sphere because the sphericity of dodecahedron is close to 1, but there are many edges on the surface. The relative erosion rate is defined as the ratio of the erosion rate of polyhedron

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to the erosion rate of sphere. With the increase in the particle sphericity, the erosion rate decreases first and then increases as shown in Fig. 20. The erosion rate is the

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lowest when the sphericity is equal to 0.77. In order to analyze quantitatively the relationship between erosion rate and sphericity, the regression equation of sphericity

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is adopted (Fig. 20). The fitting parameters of the cubic polynomial function are listed. The residual sum of squares is equal to 2.96584E-5, indicating that the regression

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equation can effectively reflect the influence of sphericity on erosion rate. As mentioned above, the erosion rate is mainly affected by the impact velocity,

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impact angle and impact concentration in addition to materials of particles and wall. The fitting curve in Fig. 20 indicates that the erosion rate is the lowest when the

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sphericity is equal to 0.77. Therefore, the influences of various factors on erosion rate

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are shown in Fig. 21. When the sphericity is less than 0.77, the erosion rate is mainly affected by the impact velocity and impact angle; when the sphericity is greater than 0.77, the influence of impact concentration on erosion rate is more obvious.

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5 Conclusions

The CFD–DEM coupling approach was employed to investigate the erosion

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behaviors of sulfur-particle-laden gas in a 90o elbow. In order to verify the model and approach in this study, the results of a verification case were compared with experimental results. The causes for rare erosion scars were analyzed from particle trajectories and secondary flow vortices. The DEM framework was used to model different polyhedral particles and the influence of particle shapes on erosion were investigated. According to the results, the conclusions can be drawn below: Firstly, the V-shaped erosion scar is attributed to the secondary collisions between the rebound particles and wall. Two obvious red erosion scars in Area A are

ACCEPTED MANUSCRIPT caused by the effects of sliding collision and direct collision. Secondly, the rare erosion scars were observed on the side walls of the downstream straight pipe near the elbow outlet. From the inlet to the outlet of the elbow, the turbulence intensity increases near the wall. The secondary flows and vortices appear, thus causing these erosion scars.

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Thirdly, the impact concentration, impact velocity and impact angle mainly affect the erosion behaviors by polyhedral particles. With the increase in the particle

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sphericity, the erosion rate decreases firstly and then increases. When the sphericity is less than 0.77, the erosion rate is mainly affected by the impact velocity and impact

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angle; when the sphericity is greater than 0.77, the influence of impact concentration

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on erosion rate is more obvious.

Acknowledgments

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The authors acknowledge the support from the National Natural Science Foundation Project of China (No. 51374177).

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[43] S.M. El-Behery, M.H. Hamed, K.A. Ibrahim, et al. CFD Evaluation of Solid Particles Erosion in Curved Ducts, J. Fluids Eng. 132(2010) 1-10. [44] Y.I. Oka, K. Okamura, T. Yoshida. Practical estimation of erosion damage caused by solid particle impact Part 1: effects of impact parameters on a predictive equation, J. Wear. 259 (2005) 95-101. [45] 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, J. Wear. 259 (2005) 102-109. [46] J.M. Ting, B.T. Corkum, Computational Laboratory for Discrete Element Geomechanics, J.

ACCEPTED MANUSCRIPT Comput. Civ. Eng. 6 (1992) 129–146. [47] G.C. Pereira, F.J. de Souza, D.A. de Moro Martins, Numerical prediction of the erosion due to particles in elbows, J. Powder Technol. 261 (2014) 105–117. [48] H. Zhang, Y. Tan, D. Yang, F.X. Trias, S. Jiang, Y. Sheng, A. Oliva, Numerical investigation of the location of maximum erosive wear damage in elbow: Effect of slurry velocity, bend

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orientation and angle of elbow, J. Powder Technol. 217 (2012) 467–476. [49] H. Liu, Z. Zhou, M. Liu, A probability model of predicting the sand erosion profile in elbows

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for gas flow, J. Wear. 342–343 (2015) 377–390.

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Fig. 1. (a) normal displacement

δn

and tangential displacement

δt

of particle collision; (b) the

constitutive model of the interaction between two particles. Fig. 2. Schematic diagram of computational domain.

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Fig. 3. Erosion rate of extrados with different numbers of meshes: (a) 257638, (b) 415694, (c) 708564, and (d) 1136953.

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Fig. 4. The ratio of local velocity to mean velocity along centerline under different turbulent models

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Fig. 5. Influences of Saffman’s lift force, virtual mass force and pressure gradient force on the distribution of erosion rate.

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Fig.6. Comparison of calculation results of three erosion models.

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Fig.7. The standard deviation of results of three erosion models.

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Fig. 8. Comparison results between predicted erosion distribution of the elbow and experimental data from Chen et al. [32].The mass flow rate of particle is 0.000208 kg/s and the mean inlet velocity of gas is 45.72m/s. Particles have the same velocity with gas flow.

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Fig. 9. Velocity magnitude field and pressure magnitude field in the X-Z plane: (a) velocity (m/s); (b) pressure (Pa).

Fig. 10. Particle motions and positions at different time points. (a) 0.005 s; (b) 0.01 s; (c) 0.015 s;

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(d) 0.02 s; (e) 0.025 s; (f) 0.03 s. Colors represent different particle velocities.

Fig. 11. Erosion distribution of the elbow for the standard case.

Fig. 12. Trajectories of sulfur particles for the standard case. Fig. 13. (a) Particle trajectories for two chosen particles and (b) particle-free zone. Fig. 14 Contours of turbulent intensity in a cross-section normal to the elbow axis at (a) 0°, (b) 30°, (c) 60°, (d) 90°, and (f) 1D downstream of the elbow outlet.

ACCEPTED MANUSCRIPT Fig. 15. Simulated streamlines in a cross-section normal to the elbow axis at (a) 0°, (b) 30°, (c) 60°, (d) 90°, and (f) 1D downstream of the elbow outlet. Fig. 16. Velocity vectors in a cross-section normal to the elbow axis at (a) 0°, (b) 30°, (c) 60°, (d) 90°, and (f) 1D downstream of the elbow outlet.

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Fig. 17. Geometrical and DEM models for four different particles: (a) Tetrahedron; (b) Hexahedron; (c) Octahedron; (d) Dodecahedron.

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Fig. 18. Schematic diagram of the motion of four different particles on the wall: (a) tetrahedron; (b) hexahedron or octahedron; (c) dodecahedron; (d) sphere.

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Fig. 19. Influences of particle shape on erosion rate in the extrados of the elbow.

Fig. 20. Regression equation of particle sphericity.

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Fig. 21. Influences of various factors on erosion rate.

ACCEPTED MANUSCRIPT Table 1 Conditions in the standard case

Parameters

Values

Parameters of elbow

Steel 120 40 400 400 60 1.5

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Material Brinell hardness BH Diameter of pipe D (mm) Length of the horizontal pipe (mm) Length of the vertical pipe (mm) Radius of curvature of the 90° elbow R (mm) Ratio of curvature R/D

Parameters of particle

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Material Shape Density ρp (kg/m3) Diameter of particle dp (μm) Mass flow (kg/s)

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Material Density ρ (kg/m3) Kinematic viscosity coefficient (Pa·s) Mean velocity of fluid flow v0(m/s)

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Parameters of flow

Methane 0.6679 1.087e-5 20 Sulfur-solid Sphere 2046 150 0.05

ACCEPTED MANUSCRIPT Table 2 Conditions in the verification case

Parameters

Values

Parameters of elbow

Aluminum 95 25.4 100 1200 38.1 1.5

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Material Brinell hardness BH Diameter of pipe D (mm) Length of the horizontal pipe (mm) Length of the vertical pipe (mm) Radius of curvature of the 90° elbow R (mm) Ratio of curvature R/D

Parameters of particle

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Material Shape Density ρp (kg/m3) Diameter of particle dp (μm) Mass flow (kg/s)

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Material Density ρ (kg/m3) Kinematic viscosity coefficient (Pa·s) Mean velocity of fluid flow v0 (m/s)

SC

Parameters of flow

Air 1.225 1.7894e-5 45.72 Sand Sphere 2320 150 0.000208

ACCEPTED MANUSCRIPT Table 3 Properties of the particles

Hexahedron

Octahedron

Dodecahedron

Sphere



0.684

0.796

0.843

0.912

1

De (μm)

138.6

147.6

141.9

153.2

150

 p (kg/m-3)

2046

2046

2046

2046

2046

 c (kg/m-3)

1353.7

1405.2

1437.6

1454.6

RI

PT

Tetrahedron

1426.96

AC

CE

PT E

D

MA

NU

SC

De —Equivalent spherical diameter;  p —Macro-particle density;  c —Micro-particle density

31

ACCEPTED MANUSCRIPT Highlights

PT RI SC NU MA D



PT E

 

CE



A new erosion model was developed for sulfur-particle-laden gas flow and verified. A rare erosion scar on the side walls of pipe near elbow outlet is revealed and investigated. The influences of sulfur particle shapes on erosion was explored The regression equation between sphericity and erosion rate was constructed. The main factors of erosion rate vary with particle sphericity.

AC



32

Graphics Abstract

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