Numerical prediction of erosion distributions and solid particle trajectories in elbows for gas–solid flow

Journal of Natural Gas Science and Engineering 30 (2016) 455e470

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Numerical prediction of erosion distributions and solid particle trajectories in elbows for gasesolid flow Wenshan Peng, Xuewen Cao* College of Pipeline and Civil Engineering, China University of Petroleum, No.66, Changjiang West Road, Qingdao 266580, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 November 2015 Received in revised form 7 February 2016 Accepted 13 February 2016 Available online 17 February 2016

Erosion caused by particles in pipe bends is a serious problem in the oil and gas industry, which may cause equipment malfunction and even failure. The majority of this work studies the particle trajectories and erosion distributions in pipe bends under different influencing factors by using the computational fluid dynamics (CFD) method. A two-way coupled Eulerian-Lagrangian approach is employed to solve the gasesolid flow in the pipe bend. Eight commonly used erosion models and two particle-wall rebound models are combined to predict the erosion rate on the 90
elbows. The Det Norske Veritas (DNV) erosion model with the Forder et al. particle-wall rebound model is finally chosen as a sample to develop the new CFD-based erosion model after comparing with the experimental data. The accuracy of this presented model is assessed by the experimental data available in previous literature for a range of flow conditions. Good agreement between the predictions and experimental data is observed. Further, the erosion distributions and particle trajectories in pipe bends under different flow velocity, particle mass flow rate and mean curvature radius to diameter (R/D) ratio and pipe diameter are investigated by applying the presented model. The results show that totally two types of erosion scars and three types of particle collisions occur at the elbows with different erosion parameters. These two types of scars may occur alone or occur together due to the combined effect of the particle collisions. Finally, two equations for predicting the maximum erosion location are obtained considering the pipe bend orientation, the particle diameter and the R/ D ratio. © 2016 Elsevier B.V. All rights reserved.

Keywords: Erosion prediction DNV erosion model with the Forder et al. particle-wall rebound model CFD Gas-solid flow Two-way coupling

1. Introduction Production of sand in gas pipelines is one of the major causes of the erosion. As a common component in the oil and gas transportation system, the 90
elbow is always used to change the flow direction. Abrupt diversion in the flow direction can cause great change in the distribution of solid particles. Studies show that the mass loss due to erosion on elbows is about fifty times larger than that on straight pipelines (Lin et al., 2015). The erosion of elbows may result in equipment malfunction and even failure, which can cause oil or gas spill and some other environmental disasters (Zhang et al., 2007; Mazumder, 2007; Najmi et al., 2015). Therefore, obtaining effective methods to

* Corresponding author. E-mail address: [email protected] (X. Cao). http://dx.doi.org/10.1016/j.jngse.2016.02.008 1875-5100/© 2016 Elsevier B.V. All rights reserved.

predict the erosion distribution of the elbow is essential for saving maintenance time and resources. Additionally, the accurate erosion prediction makes it easier to find the severe erosion locations and evaluate the service life of pipelines (Pereira et al., 2014). A lot of experimental efforts have been devoted to investigating the particle erosion for gasesolid flow in elbows. Unfortunately, most of the experiments were only conducted to study the maximum erosion rate of the whole elbow or the linear erosion profile around the elbow (Bikbaev et al., 1972 1973b; Eyler, 1987; Chen et al., 2004). Only a few papers were found that attempted to give more information of the three-dimensional erosion profiles. Kesana et al. (2013) fixed 16 ultrasonic transducers at different locations on the outer surface of the elbow to investigate the erosion distributions of the elbow. Although more erosion locations would be found, it was inadequate for providing a detailed threedimensional map of erosion scars. Solnordal et al. (2015) used a Sheffield Discovery Ⅱ D-8 co-ordinate measurement machine

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(CMM) to map out three-dimensional erosion profiles of the elbow. In their work, the erosion data of 38 lines along the flow direction in the inner surface of the elbow were measured. The erosion profiles were similar and all presented an elliptical erosion scar. The other types of erosion scar were not found in his experiment due to the defined experimental conditions. It is difficult to obtain the erosion distributions regarding the effects of all influencing factors on erosion through experiments, as they are numerous. It is more efficient to use CFD method to study a broad range of flow conditions through elbows. Edwards et al. (2001) analyzed two 90
elbows with R/D ratios of 1.5 and 5 using the Ahlert erosion model. Only elliptical erosion scars were found in these elbows. The causes of these scars were simply attributed to the roughly particle impingements without further detailed analysis. Chen et al. (2004) studied the gasesolid erosion distribution and particle trajectories in a standard elbow. They used the Ahlert erosion model and two different particle-wall rebound models to track the particles and calculate the erosion rate of elbows. The vee-shaped erosion scar found in their work was attributed to the secondary impingements of the particles. Pereira et al. (2014) used the Oka et al. erosion model and Grant and Tabakoff particle-wall rebound model to calculate the erosion rate of a 90
elbow. The vee-shaped erosion scars were found on these elbows. The sliding collisions of the particles at lower velocities were regarded as the main cause of these scars. A further study based on Pereira et al.’s work was carried out by Duarte et al. (2015). They found that the effects of mass loading and coupling methods can significantly change the erosion profiles of the elbows. For example, the vee-shaped erosion scar would occur by using the one-way coupling method, but it became not obvious when using the four-way coupling. Additionally, as the mass loadings increased, the vee-shaped scar calculated by the four-way coupling method became increasingly fuzzy. These behaviors were attributed to the particle-to-particle collisions. Solnordal et al. (2015) found that the vee-shaped erosion scars were in line with the particles travel directions for elbows with smooth pipe wall. Although there are different studies focused on the relationship between erosion scars and particle trajectories, the generation mechanism of erosion scars have not been fully understood due to the difference of particle movement in pipe bends. Moreover, most of the erosion models are in complex forms and the prediction results only yield close agreement with defined experimental conditions. Thus, the purpose of this study is to propose a more simple erosion model for predicting the erosion distributions and particle trajectories in elbows under different influencing factors. A two-way coupled Eulerian-Lagrangian method is employed to solve the gasesolid flow in the elbow. The particle trajectories are calculated by the Discrete Radom Walk (DRW) model and the particleeeddy interaction method. Eight erosion models and two particle-wall rebound models are evaluated to determine the most suitable model for this work. The erosion profiles and particle trajectories in elbows under different flow velocity, particle mass flow rate and mean curvature radius to diameter (R/D) ratio and pipe diameter are investigated by the presented model. Finally, the prediction equations for maximum erosion location are proposed by taking the effects of pipe bend orientation, particle diameter and R/D ratio into consideration. 2. Simulation modeling The CFD-based erosion modeling includes three main steps: the

continuous phase flow field simulation, particle tracking and erosion calculation. The gas is treated as a continuous phase and solved by the NaviereStokes equations. The particles are treated as a discrete phase and solved by Newton's second law. Additionally, two-way coupling is applied between the continuous phase and discrete phase.

2.1. The continuous phase model The NaviereStokes equations are employed in this section. The general equations of continuity and momentum are given as:


 vr ! þV ru ¼ 0 vt

(1)



¼  v
! !! ! ! r u þ V$ r u u ¼ VP þ V$ t þ r g þ S M vt

(2)

! where r is the gas density, u is the instantaneous velocity vector of ¼ ! the gas, P is the static pressure, t is the stress tensor, r g is the body ! force, S M is the added momentum due to the discrete phase. The stress tensor is given as: ¼

t¼m




  2 ! ! !T  V$ u I Vu þVu 3

(3)

where m is the gas viscosity, I is the unit tensor. The standard k-ε turbulence model with standard wall function is used in this work to resolve the flow turbulence, the equations are given as:

vðrkÞ vðrkui Þ v þ ¼ vt vxi vxj vðrεÞ vðrεui Þ v þ ¼ vt vxi vxj

"

"

m mþ t sk

m mþ t sε





# vk þ Gk  rε þ Sk vxj

(4)

# vε ε ε2 þ C1ε Gk  C2ε r þ Sε vxj k k (5)

where Gk is the generation of turbulence kinetic energy due to the mean velocity gradients, ui is the velocity component in i direction, xi and xj are the spatial coordinates, sk and sε are the turbulent Prandtl numbers for k and ε, C1ε and C2ε are constants, Sk and Sε are 2 source terms, mt ¼ rCm kε , sk ¼ 1.0,sε ¼ 1.3, C1ε ¼ 1.44, C2ε ¼ 1.92, Cm ¼ 0.09.

2.2. The dispersed phase model The particle trajectories are acquired by integrating the motion equation of the particles under the Lagrangian coordinates. The governing equation of particle motion is proposed according to Newton's second law:

! ! ! ! dup ! ¼ F D þ F P þ F VM þ F G dt

(6)

From the first term to the fourth term on the right hand of Eq. (6): the drag force, the pressure gradient force and the added mass force and the buoyancy force. The drag force is the main hydrodynamic force that acts on the particles:

W. Peng, X. Cao / Journal of Natural Gas Science and Engineering 30 (2016) 455e470

! 18m Cd Rep
! ! u  up FD¼ rp d2p 24

(7)

! where u p is the particle velocity vector, dp is the particle diameter, rp is the density of particles, Rep is the particle Reynolds number:

Rep ¼

! ! rdp up  u m

(8)

Cd is the drag coefficient:

Cd ¼ a1 þ

a2 a þ 3 Rep Re2p

(9)

where a1, a2, a3 are constants for the smooth spherical particles. These three parameters vary with the Reynolds number, which are given by Morsi and Alexander (1972). The pressure gradient force is caused by the pressure change in the flow:

! FP ¼

! r VP rp

(10)

457

DRW model, using the instantaneous fluid velocity to integrate the trajectory as follows:

u ¼ u þ u0 ðtÞ

(14)

The turbulent fluctuating velocity that follows a Gaussian distribution is given by: 0

u ¼2

qffiffiffiffiffiffiffi 0 u2

(15)

where z is a random number obeys normal distribution. Assuming the local turbulence is isotropic, and then the local root mean square (RMS) value of the velocity fluctuation can be calculated by:

qffiffiffiffiffiffiffi rffiffiffiffiffiffi 2k 0 u2 ¼ 3

(16)

Paticle can damp or produce turbulent eddies. Particle damping and turbulence eddies can change the turbulent quantities. The particle source terms are added in Eqs. (4) and (5) to take this effect into account. The turbulent kinetic energy of gas phase is modified by the formulation described by Faeth (1986) and Amsden et al. (1989).

The virtual mass force is given as:

! F VM


! ! 1 rd u  up ¼ 2 rp dt

2.4. Erosion models

(11)

The buoyancy force is given as:

 rp  r ! ! FG¼ g rp

(12)

Since the particles in this work are small, the pressure change over a distance of a particle diameter is negligible. Meanwhile, as the density of the fluid is much lower than the density of the particles, the pressure gradient force can be neglected. As the virtual mass force is important only when the fluid density is larger than the particles density, the virtual mass force can also be neglected. 2.3. Coupling between the two phases To obtain accurate particle trajectories and erosion distributions, the coupling between the continuous phase and the dispersed phase is also needed to be considered, especially in the conditions that the particle mass loading rate is high or the particle collision is intense (Lin et al., 2014). 2.3.1. Momentum coupling The momentum exchange is computed by examining the change of the particle momentum when it passes through each control volume, which is expressed as:

SM ¼

X

ðFD þ FG Þmp Dt

(13)

2.4.1. DNV erosion model Det Norske Veritas (DNV) (2007) developed an erosion model for predicting the erosion of straight pipes, elbows, plugged tees, welded joints and reducers. This model was developed based on numerous experimental data and numerical predicted results. The model is expressed by:

ER ¼ CFðqÞVPn FðqÞ ¼

ð  1Þiþ1 Ai qi

(18)

i¼1

where ER is the erosion rate of the target, it is defined as the wall mass loss per unit area and per unit time. In this study, ER is finally given as the penetration rate, which is evaluated by dividing the erosion rate by the density of pipe wall material. F (q) is the impact angle function, C is a constant, for steel pipes, C ¼ 2.0  109, n is the velocity exponent, for steel pipes, n ¼ 2.6. The values of Ai are given in Table 1. 2.4.2. E/CRC erosion model This model was proposed by the Erosion/Corrosion Research Center (E/CRC) in the University of Tulsa (Ahlert, 1994; McLaury, 1996). The model is based on many direct impact experiments for different impact angles and particle shapes with the purpose of predicting the erosion of carbon steel with dry or wet surface. The equations are given by Zhang et al. (2007):

ER ¼ CðBHÞ0:59 FS unP FðqÞ

where mp is the particles mass flow rate, Dt is the time step. 2.3.2. Turbulence coupling The stochastic tracking approach is used to predict the effect of turbulent flow fluctuations on particle trajectories. The dispersion of particles caused by the gas phase turbulence is calculated by the

8 X

(17)

FðqÞ ¼

5 X

Ri qi

(19)

(20)

i¼1

where C ¼ 2.17  107 for carbon steel, BH is the Brinell hardness of the target material, n ¼ 2.41, Fs is the particle sharpness factor,

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W. Peng, X. Cao / Journal of Natural Gas Science and Engineering 30 (2016) 455e470

Fs ¼ 1 for sharp sand particles, 0.53 for the semi-rounded sand particles, and 0.2 for the rounded sand particles. The values of Ri are given in Table 1. 2.4.3. Neilson and Gilchrist erosion model Neilson and Gilchrist (1968) developed an erosion model based on their experimental results. They proposed two equations for predicting the erosion rate at small and large impact angles. In the following equations, the first term on the right hand side represents erosion caused by cutting, and the second term represents erosion caused by deformation.

ER ¼

ER ¼

pq u2p cos2 q sin 2q 0

2εC u2p cos2 q 2εC

þ

þ

u2p sin2 q

u2p sin2 q 2εD

q < q0

2εD

(21)

q > q0

(22)

where q0 is the transition angle, normally set as p/4,εC is the cutting coefficient, set as 3.332  107, εD is the deformation coefficient, set as 7.742  107. 2.4.4. Oka et al. erosion model Oka and Yoshida (2005), Oka et al. (2005) took more influencing factors into account than the above three models and proposed a model as follows:

ER ¼ 1:0  109 rw kFðqÞðHvÞk1 n1


u k2 d k3 p

V

p

0

d

(23)

0

n2

FðqÞ ¼ ðsin qÞ ½1 þ Hvð1  sin qÞ

(24)

where rw is the density of target material, Hv is the Vickers hardness of the target material, dp is the particle diameter, d’ is the reference diameter, V0 is the impact velocity of the reference particle. k, k1,k2,k3,n1,n2,d'and V0 are listed in Table 1. 2.4.5. Ahlert erosion model Ahlert (1994) developed a model to predict erosion for AISI1018 steel. According to his study, the erosion model is given by:

ER ¼ AðBHÞ0:59 FS unP FðqÞ

(25)

FðqÞ ¼ aq2 þ bq

q  q0

FðqÞ ¼ x cos2 q sinðwqÞ þ y sin2 q þ z

(26) q > q0

(27)

where A is a constant, A ¼ 15.59  107 for carbon steel, q0 is the transition angle, normally set as p/12,n ¼ 1.73. a,b,w,x,y and z are listed in Table 1.

2.4.6. Gnanavelu et al. erosion model Gnanavelu et al. (2011) proposed a model to predict the erosion rate of various geometries caused by slurry erosion. The model was developed based on the experimental data of the material erosion and CFD simulations.

h i ER ¼ u2p Aðsin qÞ4 þ Bðsin qÞ3 þ Cðsin qÞ2 þ Dðsin qÞ þ E  F (28) where A, B, C, D, E and F are all correlation coefficients which are listed in Table 1.

2.4.7. Hashish erosion model Hashish (1987) developed an erosion model based on the summation of cutting and deformation erosion. The model is expressed in Eq. (29):

0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n pffiffiffiffiffiffiffiffiffiffi rp 100 3 @ A sin 2 q sin q ER ¼ pffiffiffiffiffiffidp up 0:6 2 29 3sRf

(29)

where rp is the density of the solid particle, n ¼ 2.54, s is the plastic flow stress, Rf is the roundness factor.

2.4.8. Huang et al. erosion model Huang et al. (2008) proposed an erosion model for a single particle. In this model, the deformation damage and cutting are taken into account. 2:375 ER ¼ Kr0:1875 d0:5 ðcos qÞ2 ðsin qÞ0:375 p p up

(30)

where K is a constant depends on the target material property.

Table 1 Values of parameters in solid particle erosion erosions. DNV A1 9.370 E/CRC R1 5.3983 Oka et al. k 65 Ahlert a 38.4 Gnanavelu et al. A 0.396

A2 42.295

A3 110.864

A4 175.804

A5 170.137

A6 98.398

A7 31.211

A8 4.170

R2 10.1068

R3 10.9327

R4 6.3283

R5 1.4234

k1 0.12

k2 2.3(Hv)0.038

k3 0.19

n1 0.71(Hv)0.14

n2 2.4(Hv)0.94

V’/(m$s1) 104

d’/um 326

b 22.7

w 1

x 3.147

y 0.3609

z 2.532

B 8.38

C 16.92

D 10.747

E 1.765

F 0.434

W. Peng, X. Cao / Journal of Natural Gas Science and Engineering 30 (2016) 455e470

2.5. Particle-wall interaction behavior In CFD software, particle-wall rebound model is always used with erosion model together to calculate the dynamic particle movement, the erosion rate and the maximum erosion location. Various restitution coefficients have been proposed to describe the effect of particle-wall rebound behavior (Grant and Tabakoff, 1975; Forder et al., 1998; Sommerfeld and Huber, 1999). In this study, the Grant and Tabakoff particle-wall rebound model and the Forder et al. particle-wall rebound model are used with the erosion prediction models to track particles and calculate the erosion. The restitution coefficient en in the normal direction and the restitution coefficient et in the tangential direction represent the change in particle velocity after impacting the wall. The model developed by Forder et al. is given as:

en ¼ 0:988  0:78q þ 0:19q2  0:024q3 þ 0:027q4

(31)

et ¼ 1  0:78q þ 0:84q2  0:21q3 þ 0:028q4  0:022q5

(32)

The model developed by Grant and Tabakoff is given as:

en ¼ 0:993  1:76q þ 1:56q2  0:49q3

(33)

et ¼ 0:988  1:66q þ 2:11q2  0:67q3

(34)

3. CFD modeling

459

resulting volume mesh. Gradual refinement is necessary for the near-wall region where high velocity gradients and boundary layer are present. A structured grid is used to mesh the surface of the cross-section. Finally, the hexahedral structured mesh is adopted to mesh the whole volume. The grid number used in this case is approximately 710,000. 3.3. Boundary conditions The solid particles are injected uniformly at the pipe inlet at the same speed as the gas. All of the injected particles are spherical. The roughness parameter is set as 0 which means the domain walls are considered to be perfectly smooth. The roughness constant is set as the default value of 0.5. Additionally, the turbulence intensity is set as 5%. 3.4. Numerical procedure The SIMPLE algorithm is used to couple pressure and velocity in order to improve convergence. The standard discretization schemes are employed for the pressure terms and the second order upwind discretization schemes are employed for the convection terms and divergence terms. The convergent criteria for all calculations are set as that the residual in the control volume for each equation is smaller than 105 or the number of iterations reaches to 5000 in the case of steady simulation. Moreover, the number of continuous phase iterations per discrete phase model (DPM) iteration is set as 5. A total of 9010 particles will be tracked in the simulation.

3.1. Case description 4. Results and discussion The commercial software ANSYS FLUENT is adopted to conduct the numerical simulations. A database of the experiment conducted by Eyler (1987) is employed in this work to investigate the performed erosion models. Eyler (1987) studied the erosion of long radius elbows in a pneumatic transport system. The test piece was a 90
elbow with a diameter of 41 mm and a curvature radius of 133.25 mm, which is shown in Table 2. Seven groups of upstream straight pipe length are evaluated in order to better represent the flow at the specimen location. The grid numbers and erosion rate of the elbow with different upstream pipe length are listed in Table 3. The maximum erosion rate of the elbow first changes as the length increases, and finally stabilizes at about 34.7 nm/s when the straight pipe length reaches 20D.Thus, in this work, a 820 mm(20D) vertical pipe upstream and a 410 mm(10D) horizontal pipe downstream of the elbow is used, which can be seen in Fig. 1. 3.2. Computational mesh Fig. 2 shows the 3-D computational mesh used in the simulation. Meshing consists of two parts: the first part is surface meshing and the second part is volume meshing. The surface mesh was carefully generated due to its important effect on the quality of Table 2 Flow conditions in Eyler's experiment. Name

Value

Fluid Velocity Particle diameter Particle density Particle mass flow rate Brinell hardness (BH)of pipe wall Pipe material density

Air 25.24 m/s 100 mm 2650 kg/m3 0.0286 kg/s 120 7800 kg/m3

4.1. An erosion prediction model for the steel elbow 4.1.1. Comparison of the impact angle functions The severity of erosion is determined by the fluid characteristics, the solid particle property and some other important parameters. Each erosion model defines a few parameters to calculate the erosion rate, such as particle velocity, particle size, pipe wall materials, fluid properties, etc. The impact angle function is one of the most important parameters among the influencing factors. Many investigators even observed that the erosion rate is also an impact angle function (Parsi et al., 2014). Since most of the proposed impact angle functions are empirical and only valid for defined situations, obtaining an appropriate impact angle function is essential to accurately predict the erosion rate. Therefore, the angle functions of the eight erosion models are compared with each other to determine which are suitable for this work. All of the solid particle erosion models used in this work can be written in the following general form:

ER ¼ HFðqÞVpn

(35)

where H is a coefficient defined by different erosion models. Fig. 3 shows the impact angle functions of eight erosion models. All of the impact angle functions are positive except the angle function of Gnanavelu et al. erosion model. As the function value cannot be negative, the Gnanavelu et al. erosion model will not be used in this paper. Fig. 3 also depicts that at the impact angle of 90
, the impact angle functions of erosion models such as DNV, E/CRC, Oka et al., Neilson and Gilchrist and Ahlert are positive, while the values of angle functions developed by Hashish and Huang et al. are equal to 0. It suggests that the impact angle function proposed by Hashish and Huang et al. are not in accordance with the performance of the ductile materials. Since the elbows analyzed in this

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Table 3 Effect of the length of upstream straight pipe on erosion rate. Upstream straight pipe length (  D)

10

12

16

20

24

30

40

Grid number Maximum erosion rate(nm/s)

503,659 38.59

545,105 36.54

627,997 33.33

709,988 34.74

792,880 34.83

917,218 34.98

1,123,547 34.78

Fig. 3. Functions of impact angle.

experimental data. Thus, the Ahlert erosion model is not suitable for this work, either. In conclusion, only four erosion models are chosen to be used in the next section, namely, the DNV erosion model, the E/CRC erosion model, the Neilson and Gilchrist erosion model and the Oka et al. erosion model.

Fig. 1. Computational geometry used in the simulation.

work are steel elbows, the values of angle functions are always positive. Hence, these two models are also excluded from the willbe-used erosion models. The function value of Ahlert erosion model is much larger than the other erosion models. Some studies (Chen et al., 2004; Pereira et al., 2014) also found that the Ahlert erosion analysis would result in much higher erosion rate compared to the

Fig. 2. Computational mesh used in the simulation.

4.1.2. Comparison between the four erosion models Fig. 4 shows the comparison of the predicted and the experimental erosion rate along elbow curvature angle. Several statistical parameters are used in this section for comparison of the numerical predictions and the experimental data. The statistical parameters are expressed as follows: Average percent error:

E1 ¼

n 1X pe n i¼1 i

where. pei ¼

(36)






ERpredicted ERexp erimental ERexp erimental i

 100

Fig. 4. Comparison between the predicted and the experimental erosion rate along elbow curvature angle.

W. Peng, X. Cao / Journal of Natural Gas Science and Engineering 30 (2016) 455e470

461

Table 4 Statistical results of erosion models for experimental data from Eyler (1987). Erosion model

E1

E2

E3

DNV-Forder et al. DNV-Grant and Tabakoff E/CRC- Forder et al. E/CRC- Grant and Tabakoff Neilson and Gilchrist- Forder et al. Neilson and Gilchrist e Grant and Tabakoff Oka et al.- Forder et al. Oka et al.-Grant and Tabakoff

21.60 45.52 42.86 56.66 45.83 60.74 119.91 40.41

50.95 46.31 51.08 56.66 57.76 60.74 147.51 79.09

63.47 36.02 43.08 31.05 45.45 26.01 164.30 91.29

the particle, the predicted result is not very accurate in this work. This might be due to the fact that the solid particles used in their experiments are much larger than those used in Eyler's experiment. Since the coefficient k3 in Eq. (23) is very small, the Oka et al. erosion model always obtains over-predicted results when calculating the erosion rate of small particles. Table 4 also suggests that the other three models give a better fit to the experimental data than the Oka et al. erosion model. More validation is needed in order to determine the most accurate model for the following studies.

Fig. 5. Comparison between the predictions using different particle-wall rebound models and the experimental data.

Absolute average percent error:

n 1X E2 ¼ pei n i¼1

(37)

Percent standard deviation:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u 1 X E3 ¼ t ðpei  E1 Þ2 n  1 i¼1

(38)

The 12 erosion data from Eyler (1987) in Fig. 4 are used to calculate the statistical parameters of the eight erosion models. The statistical results of erosion models are exhibited in Table 4. It can be noticed that the Oka et al. erosion model causes much higher error compared to the other models. Although the Oka et al. erosion model takes more influencing factors into consideration than the other models, such as the solid particle hardness and diameter of

4.1.3. Comparison between the particle-wall rebound models As the particle-wall rebound model stands for the velocity change of the post-collision solid particles, it plays an important role in predicting the particle trajectories. Experimental data from different researchers are used to evaluate the three erosion models and two particle-wall rebound models. The data used in this section are based on the experiments conducted by Bourgoyne (1989), Evans et al. (2004), and Pyboyina (2006). Experimental conditions are summarized in Table 5, in which u0 is the initial gas velocity. Fig. 5 shows the comparison between the predicted and the experimental erosion rate with different particle-wall rebound models. The statistical results of erosion models are shown in Table 6. It shows that the erosion rate calculated by the DNV erosion model and Forder et al. particle-wall rebound model produces the most accurate results. Moreover, the results calculated by Grant and Tabakoff model provide lower values than those of Forder et al. model. This may be attributed to the different materials that they used in their experiments. Grant and Tabakoff (1975) provided the restitution coefficient by using the aluminum material as the target material, while Forder et al. (1998) used the steel as the target material to provide the coefficients. 4.1.4. New prediction model for erosion Since the prediction of the DNV erosion model combined with the Forder et al. particle-wall rebound model presents the best precision among all models, we proposed the model based on the DNV erosion model. The function of impact angle is developed by fitting a large number of numerical simulation results. The new

Table 5 Condition of the cases. Fitting type

Fluid

D(mm)

R/D

u0(m/s)

dp(mm)

rp (kg/m3)

mp (kg/s)

rw (kg/m3)

BH

Bourgoyne

Air Air Air Air Air Air Natural gas Natural gas Natural gas Air Air Air

52.5 52.5 52.5 52.5 52.5 52.5 101.6 101.6 101.6 50.8 50.8 50.8

1.5 1.5 1.5 1.5 1.5 5 5 5 5 1.5 1.5 1.5

32 47 72 98 93 108 23 35 63 12.2 18.9 28

350 350 350 350 350 350 150 150 150 150 150 150

2650 2650 2650 2650 2650 2650 2650 2650 2650 2650 2650 2650

0.04611 0.06758 0.1182 0.1198 0.1306 0.1410 0.000148 0.000188 0.000271 0.00017 0.000521 0.000917

7800 7800 7800 7800 7800 7800 7800 7800 7800 7800 7800 7800

120 120 120 120 120 120 160 160 160 160 160 160

Evans et al.

Pyboyina

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W. Peng, X. Cao / Journal of Natural Gas Science and Engineering 30 (2016) 455e470

Table 6 Statistical results of erosion models for experimental data from Bourgoyne(1989), Evans et al.(2004) and Pyboyina (2006). Erosion model

E1

E2

E3

DNV-Forder et al. DNV-Grant and Tabakoff E/CRC- Forder et al. E/CRC- Grant and Tabakoff Neilson and Gilchrist- Forder et al. Neilson and Gilchrist e Grant and Tabakoff

5.26 31.26 37.60 52.85 45.93 56.85

19.94 32.54 38.25 52.85 48.67 56.85

30.83 23.10 22.81 17.22 34.01 27.64

4.2. Erosion distributions and particle trajectories Table 7 Parameter values in the new erosion model. B1

B2

B3

B4

B5

B6

B7

B8

11.620

62.571

181.297

300.133

291.035

163.078

48.633

5.931

model is given as:

ER ¼ 2  109 f ðqÞu2:6 p

f ðqÞ ¼

8 X

ð  1Þiþ1 Bi qi

(39)

(40)

i¼1

where the values of Bi are listed in Table 7. 4.1.5. Validation of the presented model To verify the accuracy of the presented erosion model, the erosion rate of the elbows under different parameters are calculated and compared with the previous experimental data. An erosion deviation coefficient (h) is defined to study the relationship between erosion rate calculated by the presented erosion model and the experimental data:

h¼

ERpredicted  ERexperimental  100% ERpredicted þ ERexperimental

(41)

The accuracy of the presented erosion model is clearly demonstrated in Fig. 6. It shows that the maximum prediction error is smaller than 25% which indicates good agreement between the experimental data and predicted results over a broad range of erosion rates for gasesolid flow. The detailed information of the cases and the predicted results are given in Table 8.

Fig. 6. Correlation between the predictions using the presented erosion model and the measured erosion rate.

4.2.1. Erosion distributions and particle trajectories in pipe bend under different inlet velocity Fig. 7 shows the erosion distributions and flow field profile are similar in elbows under different inlet velocity. The maximum erosion locations of these three cases all occur at 48
, which are in good agreement with the experimental data of Bourgoyne (1989), as shown in Fig. 8. Therefore, the erosion regularities can be demonstrated by focusing on case 2 as an example. The particle trajectories drastically change at the outermost side of the elbow and particularly, the particles travel in two major directions which are aligned with the vee-shaped erosion scar, as shown in Fig. 9. This regularity is in accordance with the work of Chen et al. (2004), Pereira et al. (2014) and Solnordal et al. (2015). The vee-shaped erosion scar is attributed to the particle trajectories in the pipe bend. Fig. 10 shows that the particles trajectories are divided into two paths after the particles impacting the elbow. All particles impact the elbow directly from the inlet and cause the first collision. The particles will produce a narrow secondary collision after rebounding from the first collision region. However, the secondary collision cannot be identified clearly in Fig. 9. This is due to the fact that point b in the secondary collision (path aebec) is close to the first collision point a. As a result, the first collision and the secondary collision occur very close and they work together causing the vee-shaped erosion scar. This finding is different from the work of Pereira et al. (2014). They thought that the vee-shaped scar was caused by the sliding collision of particle at lower velocities. However, the sliding collision (path aedee) found in our study can only increase the erosion in the elbow and will not significantly influence the shape of the vee-shaped erosion scar. A particle free zone at the inner side of the elbow emerges as a consequence of these two paths. In this zone, the elbow is free of erosion, as shown in Fig. 10. The particle free zone is caused by the shadow effect which have been explained by Sommerfeld and Huber (1999). Additionally, Fig. 7 shows that secondary flow occurs at several sections of the elbow. In order to investigate the effect of secondary flow on the particles, the path lines are studied in Fig. 11. There are three main path lines, namely paths 1, 2 and 3. All of these paths occur at the innermost side of the elbow and then flow to the downstream of the pipe bend, particularly, paths 1 and 2 flow along the central line of the pipe bend while path 3 gradually flows to the outer wall of the downstream pipe. Although the secondary flow is remarkable from the side view in Fig. 11, it has little influence on the particle trajectories due to the large Stokes number. As a ratio of particle relaxation time and fluid characteristic time, Stokes number can reflect the relative magnitudes of the particle inertia force and the drag force. It is a dimensionless number for representing the curvilinear movement of the solid particle. The form of this number is St ¼ rp d2p u=ð18mDÞ.When St < 1, the solid particles have a good following property with the surrounding fluid, and when St>>1, the fluid has little effect on motion of the particles. For case 2, the Stokes number is equal to 900, much larger than 1. Thus, the influence from the gas phase secondary flow is negligible. The secondary flow first appears at the top of 60
section, then flow along the outer edge of

W. Peng, X. Cao / Journal of Natural Gas Science and Engineering 30 (2016) 455e470

463

Table 8 Comparison of experimental data with predictions for gasesolid flow. Fitting type

Case

Fluid

D(mm)

R/D

u0(m/s)

dp(mm)

mp (kg/s)

BH

Measured erosion(m/s)

Present(m/s)

h(%)

Eyler Bourgoyne

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Air Natural Gas Natural Gas Natural Gas Air Air Air Air Air Air Air Air Air

41 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 52.5 152 203 254 304 101.6 101.6 101.6 50 50 50 50 52.5 52.5 50.8 50.8 50.8

3.25 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.875 2.875 2.875 2.875 4.5 1.5 1.5 1.5 1.5 1.5 3.5 3.5 3.5 3.5 5 5 5 2.4 3.6 4.2 7.8 1.5 1.5 1.5 1.5 1.5

25.24 47 72 93 98 98 103 167 169 177 177 178 203 205 222 108 109 108 104 108 108 107 111 107 106 103 141 107 141 107 111 100 100 100 100 100 100 100 100 100 23 35 63 50 50 50 50 21.35 30.5 12.2 18.9 28

100 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 150 150 150 295 295 295 295 300 300 150 150 150

0.0286 0.06758 0.1182 0.1306 0.11978 0.13939 0.14098 0.20458 0.25016 0.3498 0.2915 0.2889 0.2968 0.3816 0.3021 0.05009 0.09249 0.096195 0.15317 0.17119 0.20776 0.2968 0.38425 0.60155 0.636 0.7473 0.1895 0.4081 0.1736 0.3207 0.4028 0.106 0.186 0.305 0.636 0.768 2.65 2.65 2.65 2.65 0.000148 0.000188 0.000271 0.3366 0.3366 0.3366 0.3366 0.01715 0.02466 0.00017 0.000521 0.000917

120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 140 140 140 140 140 120 120 120 120 120 120 120 120 120 160 160 160 120 120 120 120 109 109 160 160 160

2.86E-08 3.32E-07 1.65E-06 3.70E-06 4.23E-06 4.94E-06 5.29E-06 3.73E-05 4.77E-05 8.33E-05 7.38E-05 6.52E-05 7.76E-05 7.96E-05 7.01E-05 3.56E-06 5.64E-06 5.29E-06 9.88E-06 1.38E-05 1.37E-05 1.43E-05 2.62E-05 3.56E-05 3.26E-05 2.96E-05 3.32E-05 4.94E-05 3.02E-05 4.35E-05 2.12E-05 4.00E-06 8.00E-06 1.30E-05 2.85E-05 2.80E-05 9.50E-06 5.40E-06 3.60E-06 2.40E-06 1.72E-11 9.04E-11 6.80E-10 4.66E-06 3.97E-06 3.28E-06 2.42E-06 5.16E-09 1.82E-08 2.04E-11 1.82E-10 7.06E-10

3.27E-08 3.41E-07 1.80E-06 3.93E-06 4.17E-06 4.81E-06 5.69E-06 3.21E-05 3.99E-05 6.08E-05 5.33E-05 5.19E-05 7.42E-05 9.41E-05 9.03E-05 2.55E-06 5.15E-06 5.04E-06 7.99E-06 8.98E-06 9.69E-06 1.32E-05 1.94E-05 2.79E-05 2.85E-05 2.95E-05 2.27E-05 2.95E-05 1.97E-05 2.64E-05 1.71E-05 3.95E-06 6.96E-06 1.15E-05 2.54E-05 2.77E-05 9.50E-06 5.43E-06 3.35E-06 2.40E-06 1.83E-11 9.01E-11 6.74E-10 2.86E-06 2.68E-06 2.15E-06 1.60E-06 8.53E-09 3.02E-08 1.94E-11 1.95E-10 8.67E-10

6.68 1.36 4.30 3.01 0.75 1.36 3.66 7.57 8.94 15.64 16.15 11.34 2.22 8.35 12.57 16.60 4.50 2.44 10.59 21.16 17.13 4.13 14.92 12.04 6.78 0.19 18.75 25.24 21.00 24.49 10.85 0.68 6.93 6.12 5.71 0.49 0.01 0.30 3.66 0.09 3.19 0.15 0.42 23.95 19.36 20.84 20.32 24.62 24.84 2.62 3.42 10.22

Evans et al.

Bikbaev et al.

Weiner and Tolle Pyboyina

this section, and finally turn to flow along the center line, as shown in Fig. 11. Nevertheless, there are great differences in the intensity of the secondary flows under different inlet velocity, the scale of the secondary flow under 93 m/s is larger than that of 72 m/s and 47 m/ s, as shown in Fig. 12. Although the intensity of the secondary flow becomes stronger, the influence on the particle trajectories becomes weaker due to the increasing Stokes number. 4.2.2. Erosion distributions and particle trajectories in pipe bend under different particle mass flow rate The erosion distribution and the particle concentration under different particle mass flow rate is shown in Fig. 13. The erosion contours show a similar distribution regularity with the particle concentration. Fig. 14 indicates that the maximum erosion locations for cases 32e36 all appear at 48
in the elbow which shows a good fit to experimental data of Bourgoyne (1989). Additionally, the highest particle concentration locations also occur at 48
. In Fig. 15, the

erosion rata increase linearly from 3.946 mm/s to 27.728 mm/s as the particle mass flow rate increases from 0.106 kg/s to 0.768 kg/s. The particle trajectories of cases 32e36 are similar with case 2 which can be seen in Figs. 9 and 10. Only vee-shaped erosion scars are found on elbows for cases 32e36. These scars are caused by the combined effect of the three types of particle collisions which has been descried in Section 4.2.1. In particular, two high particle concentration zones occur in the downstream of the pipe bend, namely, Zone C and Zone D, as shown in Fig. 16. On one hand, these two zones are in line with the particle trajectories dramatic change region, and on the other hand, the particle trajectories can directly influence the erosion contour, as shown in Figs. 13 and 16. Thus, the particle concentration distribution can represent the erosion location to some extent. 4.2.3. Erosion distributions and particle trajectories in pipe bend under different R/D ratio The R/D ratio of the elbows for cases 41e43 is equal to 3. The

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Fig. 7. Flow field and erosion distribution of the elbows for cases 2e4.

Fig. 10. Elevation view and plan view of particle trajectories for case 2. Fig. 8. Comparison between numerical and experimental penetration rate versus bend curvature angle for cases 2e4.

maximum erosion location of elbows occurs at 29
around the elbow, as shown in Fig. 18. It is smaller than that in the elbows with an R/D ratio equal to 1.5, as shown in Fig. 17. This may be due to the fact that for the elbow with larger R/D ratio, the particles

will impact the elbow at a smaller angle. Zone A and Zone B in Fig. 18 are two special zones, with more peaks than that in Fig. 14. As the R/D ratio increases, the elbow length increases and the impact and rebound angles decrease, as a result, the particles in longer radius pipe bend experiences multiple collisions with the outer wall.

Fig. 9. Erosion contour and particle trajectories for case 2.

Fig. 11. Three views of path lines for case 2.

W. Peng, X. Cao / Journal of Natural Gas Science and Engineering 30 (2016) 455e470

465

Fig. 12. Secondary flow at the section of 60
for cases 2e4.

Compared to the erosion profiles of cases 2e4, the erosion distributions of cases 41e43 are more diverse. A grooved erosion scar emerges behind the vee-shaped erosion scar along the gas flow direction, as shown in Fig. 19. The vee-shaped scar has been detailed analysis by some researchers mentioned in Section 4.2.1. Unfortunately, very little detailed information has been provided to explain the grooved erosion scar. The grooved erosion scar occurs in Zone B that between q ¼ 70
and 90
around the elbow, as shown in Fig. 18. The grooved scar is a little far away from the veeshaped scar which indicates that there are some differences in the particle collision. Unlike the vee-shaped scar caused by the combined collision of the first and secondary collision in Fig. 9, the vee-shaped scar in Fig. 19 is attributed to the single first collision.

Fig. 14. Comparison between numerical and experimental penetration rate versus bend curvature angle for cases 32e36.

Point b’ in direct collision path (a0 eb0 ec0 ) is a little far from point a’ and this secondary collision causes the grooved erosion scar, as shown in Fig. 20. Beside the difference in the erosion scar, the area

Fig. 13. The relationship between the particle concentration and erosion rate of the elbows for cases 32e36.

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W. Peng, X. Cao / Journal of Natural Gas Science and Engineering 30 (2016) 455e470

of particle free zone in Fig. 20 is larger than that in Fig. 10. The further distance from a’ to b’ is the major contribution to this difference. Fig. 21 shows that only one path line occurs at the innermost side of the elbow and this path finally flows to the outlet. It indicates that the flow fields and the secondary flow intensity in long-radius elbows are much plain than that in shortradius elbows in Fig. 11. Hence, the gas phase has less influence on the particles movement in long-radius elbows.

Fig. 15. Predicted penetration rate and liner fit of the data for cases 32e36.

4.2.4. Erosion distributions and particle trajectories in pipe bend under different pipe diameter Pipe diameter is another important parameter that influences the erosion distribution. Fig. 22 shows the relationship between the particle trajectories and the erosion rate of the elbows for cases 37e40. The erosion rate decreases from 9.50 mm/s to 2.40 mm/s as the pipe diameter increases from 0.152 m to 0.304 m which suggests that there is a good linear relationship between the erosion rate and the value of 1/D2. This observation is in agreement with previous studies (Salama and Venkatesh, 1983; McLaury and Shirazi, 2000). The maximum erosion location in the pipe bends under different pipe diameter is limited between q ¼ 33
and 36
, with slight differences, as shown in Fig. 23. The distribution of particle free zone and erosion contour in Fig. 24 are similar to those in long-radius elbows in Fig. 20. There are two obvious discrepancies of the erosion scars in large-diameter elbows and in long-radius elbows. The first is that the ratio of erosion area to the whole elbow area in large-diameter elbows is smaller than that in long-radius elbows. The second is that the erosion scars in long-radius elbows is more remarkable than that in large-diameter elbows. This can be explained by the respond time and the particle rebound angle. The particles have relatively more time to respond to the change and impact the elbow at a smaller angle in long-radius pipe bend in Fig. 19 than that in largediameter pipe bend in Fig. 24. Fig. 24 also shows that in longradius pipe, the particle trajectories are smoother and the sliding collision is more remarkable when compared to those in largediameter pipe. 4.3. Prediction of the maximum erosion location based on the particle trajectory

Fig. 16. The relationship between the particle trajectories and the particle concentration for case 32.

In order to explain the relationship between the particle trajectory and the erosion location, the trajectory of a single particle in the pipe bend is analyzed. Fig. 25a shows the schematic of the location of elbow puncture point in the vertical-horizontal (VeH) upward direction. Since the particle collision direction is parallel to the gravity, the gravity will not influence the particle collision

Fig. 17. The relationship between the flow field and erosion rate of the elbows for cases 41e43.

W. Peng, X. Cao / Journal of Natural Gas Science and Engineering 30 (2016) 455e470

Fig. 18. Comparison between numerical and experimental penetration rate versus bend curvature angle for cases 41e43.

467

Fig. 21. Three views of path lines for case 41.

1

qmax ¼ cos

1  Rd 1 þ 12$DR

! (42)

Fig. 25b shows the schematic of the location of elbow puncture point in the horizontal-vertical (HeV) upward direction. Since the particle collision direction is perpendicular to gravity, the particles will deviate from the straight flow direction and as a result, large particles will impact the elbow wall at a lower point (point G) than that of small particles (point F). In Fig. 25b, dc is the critical diameter, d1 is the distance of small particle trajectory to the central line and d2 is the distance of large particle trajectory to the central line.

Fig. 19. Erosion contour and particle trajectories for case 41.

direction. In Fig. 25a, d is the distance from the particle trajectory to the central line of the pipe bend. The equation of the maximum erosion location (point E) due to the particle collision can be written as:

Fig. 20. Elevation view and plan view of particle trajectories for case 41.

Fig. 22. The relationship between the particle trajectories and the erosion rate of the elbows for cases 37e40.

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W. Peng, X. Cao / Journal of Natural Gas Science and Engineering 30 (2016) 455e470

Fig. 23. Comparison between numerical and experimental penetration rate versus bend curvature angle for cases 37e40.

The equation for the maximum erosion location caused by different diameter particles can be written as:

qmax

d ! 1 1 8 1 R > cos > > 1 D > > 1þ $ < 2 R ¼ > d ! > > 1 2 > > : cos1 R 1 D 1þ $ 2 R



dp < dc



Bourgoyne (1989) developed an equation based on his experimental data with a form similar to Eq. (42). In his equation, d2 is determined as 0. Although the equation of Bourgoyne (1989) is easy to use, it cannot accurately predict the erosion location of the elbows in operating conditions which are different from his experiments. El-Behery et al. (2010) also proposed an equation based on the bend geometry and his numerical predictions. The form of his equations is similar to Eq. (43), and the critical diameter in his equations is 100 mm. Moreover, d1 is determined as 3/8D and d2 is determined as 1/4D, which means that d2 is half the length from the central line to the inner wall and d1 is half the length from d2 to the inner wall. This equation takes the influence of critical diameter into consideration which makes it more accurate in predicting the maximum erosion location. For the H-V upward pipe bend, the maximum erosion location can be changed not only by the pipe geometry but also by the gravity. Since the particles will spend more time traveling in longradius elbow than that in short-radius elbow under the same velocity, the gravity can drive the particles deviate the streamline and impact the elbow at a lower angle. Remarkable difference can be found in Figs. 8 and 18, with the R/D ratio equal to 1.5 and 3 respectively. The R/D ratio has a significant influence on the maximum erosion location, but unfortunately, neither Bourgoyne's equation nor El-Behery et al.’s equation takes this influencing factor into account. In this work, we obtain a more comprehensive equation for the HeV upward pipe bend based on the form of Eq. (43), which is given as:

(43)

dp  dc

1



qmax

8 cos > > > > > > > > > > > < ¼ cos1 > > > > > > > > > > > : cos1

d1 ! R 1 D 1þ $ 2 R d ! 1 2 R 1 D 1þ $ 2 R d ! 1 3 R 1 D 1þ $ 2 R 1







dp < dc



dp  dc ; R=D < c



dp  dc ; R=D  c

(44)



In Eq. (44), c is the critical R/D ratio and d3 is the length from the large particle collision point to the central line of the long-radius elbow. d3 is the smallest among the three lengths (d1,d2,d3) due to the long-radius effect. As the calculation of the undetermined parameters in Eq. (44) is complex, further investigation is needed to determine them. 5. Conclusions There are various erosion prediction models for gasesolid flow, but most of them have complex forms and can only work well within limits. It is very essential to obtain a simple and accurate erosion model which can be more convenient implemented in CFD software to predict the erosion. In this work, a numerical erosion prediction model is developed for gasesolid flow with a simple form. Comparison of the predictions calculated by the presented erosion model with the measured erosion data show good agreement for a wide range of operating conditions. Different erosion scars and particle free zones are found and explained by the direct collision, sliding collision and shadow effect. According to the CFDbased calculation, following conclusions can be drawn:

Fig. 24. Erosion contour and particle trajectories for case 37.

(1) The particle trajectories and erosion distributions are similar for short-radius elbows (cases 2e4) under different inlet

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469

Fig. 25. The schematics of the location of elbow puncture point: (a)VeH upward, (b)HeV upward.

(2)

(3)

(4)

(5)

velocity. All of the erosion contours present vee-shaped scars. These scars are caused by the combined effect of the first direct collision and the secondary collision in path aebec, which is different from the study of Pereira et al. (2014). Additionally, the sliding collision in our study can only increase the erosion rate of the elbow, but cannot significantly change the shape of the erosion scar. The erosion rate increases linearly with the particle mass flow rate for short-radius elbows (cases 32e36). The particle trajectories and the erosion distributions for cases 32e36 are similar to the cases 2e4. Two high particle concentration zones are found in the downstream of the pipe bend. The profile of the particle concentration provides a good fit to the erosion profile, especially in the severe erosion location. The particle concentration distribution can represent the erosion location to some extent. Unlike the uniform change of the erosion rate around the elbow in short-radius elbows, the erosion rate of long-radius elbows (cases 41e43) presents many peaks. This can be attributed to the fact that the particles in longer radius pipe bend experiences multiple collisions with the outer wall. Another important finding is that the vee-shaped scar in long-radius elbow is caused by the single first collision which is different from the generation mechanism of vee-shaped scars in short-radius elbow. Moreover, a grooved erosion scar occurs a little far away from the vee-shaped erosion scar. We think that this scar is caused by the secondary collision. Since the literature about the grooved erosion scars under different operating conditions are relatively few, more efforts are needed for further investigation. There is a good linear relationship between the erosion rate and the value of 1/D2 for the elbows under different pipe diameter (cases 37e40). The vee-shaped erosion scar and the grooved erosion scar also appear in the large-diameter elbow with similar distributions as the long-radius elbow. However, the erosion scars are not as remarkable as those in longradius elbows. Additionally, the ratio of erosion area to the whole elbow area in large-diameter elbow is smaller than that in long-radius elbow. Since the values of particle Stokes numbers are very big for the cases studied in this work, the effect of secondary flow on the particle movement is negligible despite the intensity of the secondary flow in some elbows are relatively strong.

Therefore, the secondary flow drived collision cannot occur in these cases for gasesolid flow. (6) Two maximum erosion-location predicted equations are developed by taking the effects of the pipe bend orientation, the particle diameter and the R/D ratio into consideration. The location prediction have been studied preliminary and found that it has a close correlation with the R/D ratio. Unlike the equations proposed by Bourgoyne (1989) and El-Behery et al. (2010), the equations in this study detailed divide the erosion location in order to obtain more accurate prediction results. Acknowledgments The authors would like to acknowledge the support of National Natural Science Foundation of China (No.51274232) and the Fundamental Research Funds for the Central Universities of China (No.15CX06070A). References Ahlert, K.R., 1994. Effects of Particle Impingement Angle and Surface Wetting on Solid Particle Erosion of AISI 1018 Steel. Ph.D. thesis. Department of Mechanical Engineering, The University of Tulsa. Amsden, A.A., O'rourke, P.J., Butler, T.D., 1989. KIVA-II: A Computer Program for Chemically Reactive Flows with Sprays (No. LA-11560-MS). Los Alamos National Lab., NM (USA). Bikbaev, F.A., Maksimenko, M.Z., Berezin, V.L., Krasnov, V.I., Zhilinskii, I.B., 1972. Wear on branches in pneumatic conveying ducting. Chem. Pet. Eng. 8 (5), 465e466. Bikbaev, F.A., Krasnov, V.I., Maksimenko, M.Z., Berezin, V.L., Zhilinski, I.B., Otroshko, N.T., 1973. Main factors affecting gas abrasive wear of elbows in pneumatic conveying pipes. Chem. Pet. Eng. 9 (1), 73e75. Bourgoyne, A.T., 1989. Experimental study of erosion in diverter systems due to sand production. In: Proc., SPE/IADC Drilling Conference, New Orleans, LA, SPE/ IADC 18716. Chen, X., McLaury, B.S., Shirazi, S.A., 2004. Application and experimental validation of a computational fluid dynamics (CFD)-based erosion prediction model in elbows and plugged tees. Comput. Fluids 33 (10), 1251e1272. Det Norske Vertitas, 2007. Recommended Practice RP 0501: Erosive Wear in Piping Systems. Duarte, C.A.R., de Souza, F.J., dos Santos, V.F., 2015. Numerical investigation of mass loading effects on elbow erosion. Powder Technol. 283, 593e606. Edwards, J.K., McLaury, B.S., Shirazi, S.A., 2001. Modeling solid particle erosion in elbows and plugged tees. J. Energy Resour. Technol. 123 (4), 277e284. El-Behery, S.M., Hamed, M.H., Ibrahim, K.A., El-Kadi, M.A., 2010. CFD evaluation of solid particles erosion in curved ducts. J. Fluid Eng. T ASME 132 (7), 071303. Evans, T., Bennett, H., Sun, Y., Alvarez, J., Babaian-Kibala, E., Martin, J.W., 2004. Studies of Inhibition and Monitoring of Metal Loss in Gas Systems Containing Solids. Corrosion 2004. Paper No. 04362. NACE International, New Orleans. Eyler, R.L., 1987. Design and Analysis of a Pneumatic Flow Loop. M.S. thesis. West Virginia University, Morgantown, WV.

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