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Prediction of particle distribution and particle impact erosion in inclined cavities
Powder Technology 305 (2017) 562–571
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Prediction of particle distribution and particle impact erosion in inclined cavities Zhe Lin a,b,⁎, Yifan Zhang a, Yi Li a, Xiaojun Li a, Zuchao Zhu a a b
Key Laboratory of Fluid Transmission Technology of Zhejiang Province, Zhejiang Sci-Tech University, 5 Second Avenue, Xiasha Higher Education Zone, Hangzhou 310018, China Zhejiang Institute of Mechanical & Electrical Engineering Co., Ltd., Hangzhou 310011, China
a r t i c l e
i n f o
Article history: Received 7 February 2016 Received in revised form 18 September 2016 Accepted 17 October 2016 Available online 19 October 2016 Keywords: Cavity Inclination angle Solid particle erosion Computational methods
a b s t r a c t The cavity is one of the basic structures in pipe systems, but it is vulnerable in erosive environments. Cavities can occur in various places throughout industrial processes, each making a difference in the distribution of erosion. The aim of this study is to investigate gas (air)-solid flow properties and erosion characteristics in inclined cavities. A two-way URANS (SST)-DPM method was adopted. The simulations were carried out for three Stokes numbers (St), eight height differences (h) from 0 to 0.8H, and ten inclination angles from 0° to 90°. Results indicated that gas flow properties change little with inclination angle (φ). However, a variation in φ resulted in a change in the distribution of the particles and erosion on the aft wall, especially when St were 1.2 and 9.1. The particle number (N) in the zone 10 mm from the aft wall decreases linearly with φ for both previously mentioned St, and the slopes of N-φ curves increase exponentially with h. The maximum erosion rate (Em) on the aft wall also decreases linearly with φ for the St of 1.2, and the negative slope of Em-φ curve follows an exponentially increasing with h. But Em decreases quadratically with φ when St is 9.1, and the function coefficient appears a Gaussian relationship with h. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Solid particle erosion is inevitable in pipeline transportation with the working fluids containing solid particles, such as pneumatic conveying [1–3]. It is also a key factor affecting the lifetime of piping components and costs industries millions of dollars each year [4]. In past decades, to obtain erosion characteristics, a large amount of investigation has been carried out on piping components such as pipes, elbows, sudden expansion, sudden contraction, and valves. Shirazi et al. [5] developed a simple semi-empirical procedure for the estimation of erosion in piping components, and demonstrated the procedure for a tee and an elbow. Gabriel et al. [6] and Carlos et al. [7] revealed the effects of sand particle concentrations on the erosion of an elbow by simulations. Hanson et al. [8], Niu et al. [9], McLaury et al. [10,11], Akilli et al. [12], Yong Quek et al. [13] and Brown [14] also discussed the erosion in elbows by using simulation and experimental methods. Lee et al. [15] used the Eulerian approach to predict erosion in a boiler tube. Habib et al. [16–18] used the CFD method to investigate the effect of fluid flow and geometry on the erosion of a pipe contraction. Nemitallah et al. [19] indicated the effects of flow velocity, particle size, and pipe material on the downstream erosion of a sharp-edged orifice. Shabgard et al. [20] investigated the microstructural erosion ⁎ Corresponding author at: Key Laboratory of Fluid Transmission Technology of Zhejiang Province, Zhejiang Sci-Tech University, China. E-mail address: [email protected] (Z. Lin).
http://dx.doi.org/10.1016/j.powtec.2016.10.035 0032-5910/© 2016 Elsevier B.V. All rights reserved.
features and material removal mechanisms of AISI H13 core boxes using experimental methods. Nøkleberg and Søntvedt [21] developed a model, which verified with experimental results, to estimate the erosion and lifetime of chokes valves. Forder et al. [22] presented a computational fluid dynamics erosion model and predicted erosion rates within oilfield control valves. McLaury et al. [23] performed a computational study to examine the erosion in a choke geometry; they demonstrated that abrupt changes in a fitting can result in regions of high erosion. Wallace et al. [24] used a Eulerian–Lagrangian model to examine erosion in two choke valves. Cavities (Fig. 1) are one of the basic structures in pipe systems (such as in double flat gate valves). But only a few studies exist concerning particle erosion in cavities, including our own previous studies. Postlethwaite and Nesic [25] tested the erosion in an ideal cavity (where the height of the leading wall is equal to that of the aft wall) using silica sand. Wong et al. [26] investigated the erosion in a vertical ideal annular cavity (i.e., where the angle between the gravity force and the tube direction, φ, is 0°) using CFD and experimental methods. We also conducted both experimental and numerical studies investigating the characteristics of erosion in horizontal (φ = 90°) cavities. The effects of the height difference between the leading wall and aft wall on particle distribution and erosion have been analyzed [27,28]. From the above review, it is believed that previous studies have only focused on horizontal and vertical cavities. However, cavities have various orientations in process industries. Therefore, this study investigates gas-solid flow properties and erosion characteristics in inclined cavities
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in this region do not impact the distribution of particles in the area away from the bottom wall, and also have no effect on the erosion of the aft wall. For the region near the aft wall, the largest local maximum volume fraction is about 0.1% for the discussed cases. According to Hugo [33], two-way coupling is considered a reasonable approach for αp ≤ 0.1%. Therefore, particle collisions were neglected in present simulations, and two-way Eulerian-Lagrangian simulation method was used to predict the erosion of the aft wall within the cavity structure. There are three main models in the present simulation as follows: the continuous phase model used for the prediction of the carrier fluid flow-field; the particle tracking model used for the calculation of particle trajectories; and the erosion model used for the determination of degree of erosion. This study is the follow-up research of our previous work [27]. The simulation method and the simulation procedure of this study are the same as that of our previous work. Thus, the simulation method and the procedure are described briefly in this study, further details were given in our previous paper. Fig. 1. Schematic structure of the computational domain.
3.1. The continuous phase model
(Fig. 1). The Eulerian-Lagrangian simulation method is applied. The effects of the inclination angle (φ) on gas flow, particle distribution, and erosion of the aft wall are discussed.
The dynamics of incompressible flow were solved using the incompressible unsteady Reynolds-Averaged Navier-Stokes equation. The turbulence is calculated by the shear-stress transport (SST) model. 3.2. The particle tracking model
2. Problem statement The problem considered is that of solid particle erosion in an inclined cavity. In accordance with our previous study [27], a 2D computational domain is used in this paper, as shown in Fig. 1. The flow domain consists of a cavity structure and two extension tubes. The heights of the upstream tube (S) and the leading wall (H) are 40 mm and 26.7 mm, respectively. According to the upstream length function of pipe turbulent flow, and referred to the two phase studies [29,30], the tube length upstream the cavity (Lup) was set as 32S. Meanwhile, the velocity profiles at the inlet of the cavity for Lup of 32S and 150S were also compared, and the results indicated that the upstream length of 32S is reasonable for present simulations. The tube length downstream of the cavity was set as 35H. The length of the cavity (L) was three times of H. The top point of the leading wall was set as the origin of the coordinates. Discussed height differences (h) were 0, 0.1H, 0.2H, 0.3H, 0.4H, 0.5H, 0.6H, and 0.8H. The inclination angle (φ) varied from 0° to 90°. Air at 25 °C is regarded as the carrier fluid. The working pressure is assumed as 1.01 × 105 Pa. The average gas velocity at the inlet was 9.3 m/s (i.e., the centerline velocity was 10.5 m/s when the flow developed fully). As the Mach number was far b 0.3, the carrier fluid was set as incompressible. The density and the dynamic viscosity of air were set as 1.225 kg/m3 and 1.79 × 10−5 Pa s, respectively. Stokes numbers (St) directly represent the following behaviors of particles with carrier fluid. The change of St will affect the distribution of particles and particle erosions [27,28,31]. Thus, to investigate the effect of St, diameters of 15, 50, and 150 μm were employed. According to Fessler and Eaton [32], the corresponding Stokes numbers (St) are 0.12, 1.2, and 9.1 (details are presented in our previous study [27]). The density of particles is 2500 kg ⁄ m3. The investigated particle mass loading ratio is 0.2, that the volume fraction (αp) of loading particles is 0.0098% (i.e. 98 ppm by volume). The pipe material was set as carbon steel, and the Brinell hardness number was 140. 3. Simulation modeling For the investigated cases, there are two high particle density regions inside the cavity: the bottom region of the cavity and the region near the aft wall. For the bottom region, particle velocities are extremely small. So whether consider particle collisions or not, motions of particles
Particle trajectories were calculated using Newton's second law. As the Reynolds number is large in the present study, the Magnus lift force was not considered. The high value of ρp/ρ implies that the pressure gradient force and the virtual mass force were small and were therefore neglected. The Basset history force was ignored due to the low particle acceleration in simulations. By the magnitude analysis, it is found that the Saffman lift force was two orders of magnitude smaller than the drag force. Thus, the Saffman lift force on particles was also neglected. Under these assumptions, only drag and gravitation forces were used in the present calculations, shown as Eq. (1). The Random Walk Model (RWM) was also adopted for the calculation of the turbulent dispersion of particles. dV p ¼ FD þ dt
! ρ g 1− ρp
ð1Þ
where the first term on the right-hand side is the drag force, and the second term is the gravity force. Particle-wall interaction plays an important role in the calculation of particle trajectories. Hence, the coefficients of restitution (et, en) presented by Grant [34] were used to describe the collision and rebound performance between particle and wall. en ¼ 0:993−1:76θ þ 1:56θ2 −0:49θ3
ð2Þ
et ¼ 0:988−1:66θ þ 2:11θ2 −0:67θ3
ð3Þ
where θ is the impact angle of the particle. Table 1 Erosion model empirical constants for carbon steel (dry surface) and aluminum [36]. Constant
Carbon steel
Aluminum
C α a b w X Y Z
1.22 × BH−0.59 15 deg −3.34 × 10−4 1.79 × 10−4 1 1.239 × 10−5 −1.192 × 10−5 2.167 × 10−5
1.865 × 10−6 10 deg −34.79 12.3 5.205 0.147 −0.745 1
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Fig. 2. The nephogram of the streamwise velocity and the streamline of the gas phase for h = 0.
The erosion rate (mm/h), E, is calculated according to the equation of Shirazi et al. [5].
3.3. The erosion model The widely used semi-empirical single particle erosion model proposed by Ahlert [35] and McLaury [36] was adopted for the calculation of the erosion ratio. This model is relevant for the erosion study of piping components (including cavities) [26,37–39]. ER ¼ C F s W p n f ðθÞ f ðθÞ ¼
ð4Þ
E ¼ 3:6 106
s_ ER ρw Ntr A lc
ð6Þ
where ṡ denotes the sand rate (kg/s), ρw is the density of the boundary wall (kg/m3), Ntr is the number of tracking particles, A is the impingement area (m2) and ERlc is the local erosion ratio (kg/kg). 4. Simulation procedure and validation
2
aθ þ bθ Xcos2 ðθÞ sinðwθÞ þ Ysin2 ðθÞ þ Z
0≤θbα α ≤θb90 °
ð5Þ
where ER is the erosion ratio, defined as the mass eroded of the wall surface divided by the mass of the particles impacting the wall (kg/kg). Fs is the particle sharpness factor; Fs = 1 represents sharp particles, 0.53 semi-rounded particles, and 0.2 rounded particles. W is the particle impingement velocity (m/s). The value of velocity exponent n is 1.73. The empirical constants for carbon steel and aluminum are given in Table 1 [36].
The commercial software FLUENT (Ansys, Inc., Canonsburg, PA, USA) was used for numerical calculations. The velocity-inlet boundary condition (9.3 m/s) and the pressure-outlet boundary condition (1.01 × 105 Pa) were applied for the simulations. The second-order central difference scheme was applied for pressure gradient, convection terms, and divergence terms. The automated tracking scheme (switches automatically between the implicit scheme and the trapezoidal scheme) was employed for the calculation of particle motion. The gas phase and the particle phase have a same transient time-step of 0.0005 s.
Fig. 3. Gas streamwise velocities at different x positions.
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Fig. 4. Partial derivatives of streamwise velocity at different x positions.
Firstly, single gas flow over the cavity was calculated using the SIMPLEC algorithm, and this procedure ceased at a flow time of 1 s. We then input 40 uniformly distributed parcels per time-step into the flow field. For each time-step, the coupling calculation between the two phases continued until there were no more changes in particle source and gas-phase quantities between the two iterations. Meanwhile, particle impact erosions on the cavity walls for each time-step were recorded, and then erosions during 1.5–2.5 s were calculated. Structural mesh was used in the mesh modeling, and the boundary layer was refined to meet the requirements of the simulation model. A grid-independence test was performed in our previous study using 5 grid numbers (30,000, 100,000, 200,000, 360,000, and 660,000.) [27]. The result showed that more there was little change in simulation
results when the grid number is larger than 200,000. Thus, a grid number of 200,000 was used for the present calculations. The validation of our computational procedure comprises the following two parts: two-phase flow validation and erosion validation. To verify the accuracy of two-phase flow properties, the simulated streamwise mean velocities and streamwise fluctuating velocities of a vertically positioned backward-facing step were compared with the experimental data obtained by Fessler and Eaton [32]. Good agreements were achieved between our simulation results and the experimental data. For the validation of erosion, the calculation results of a vertically positioned cavity were compared with the experimental erosion rates (the depth of the eroded material per time interval) of Wong [26]. The
Fig. 5. Turbulent kinetic energies at different x positions.
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Fig. 6. The arrangement of grid cells for the calculation of particle number density.
experiments were carried out with the mass flow rate of 0.03 kg/s, and the results were recorded between different particle loadings, i.e. different time period. The 0–25 kg data set could be affected by the surface inclusions (such as the oxide layer), and the 0–300 kg data set could be influenced by the erosion morphology. Thus the 25–50 kg data set is more suitable for erosion validation. The comparison shows that our calculation erosion rates agree well with the 25–50 kg experimental data. And the average error of predictions between y/H = −0.3 and 0 is lower than 15%. Details of validations are given in our previous study [27]. 5. Results and discussion Gas-solid flow properties and erosion characteristics in inclined cavities were examined using a two-way Eulerian–Lagrangian simulation method. The effects of inclination angle on the flow of the gas phase, the motion of particles and the particle erosion were investigated. 5.1. Gas phase The flow of the gas can be reflected by the distribution of streamwise gas velocity (u), partial derivative of streamwise velocity (∂u/∂y) and turbulent kinetic energy (k). Thus, these flow characteristics for the discussed cases are extracted and compared. Comparisons show that the inclination angle has a negligible effect on the flow states of the
gas phase. This is caused by the small mass loading ratio (0.2) used in this study. Therefore, the flow characteristics for h = 0 and St = 1.2 are selected and displayed in the following section, to show the effect of φ quantitatively. When the gas passes through the ideal cavity (h = 0), a large main vortex forms in the aft part of the cavity by the shear layer (generated on the top area of the cavity), as shown in Fig. 2. And another opposite rotating vortex (secondary vortex) arose in the leading part. Meanwhile, there are two small vortexes existed at the bottom corners of the cavity. To explore the influence of φ, we extracted streamwise velocities at x = 0.05L, 0.2L, 0.5L, 0.8L and 0.95L, as plotted in Fig. 3. It is noted that the velocity curves for four inclination angles are coincident. Fig. 4 shows the partial derivatives of streamwise velocity (∂u/∂y) at previously mentioned five positions. It is clear that the shear layer region has a large ∂u/∂y, and the value decreases with the increase of x. Meanwhile, determined by the vortexes, the values of ∂u/∂y in the bottom areas of the cavity change greatly with x. By comparison, the inclination angle has little effect on ∂u/∂y. The distribution of turbulent kinetic energy presents the turbulence of gas flow. In the presence of shear layer, the turbulent kinetic energies in the top area of the cavity are larger than other positions (Fig. 5). Meanwhile, it is obvious that the distribution of k is independent on inclination angle. 5.2. Particle motion The concept of particle number density (ρpn) was employed to get a more intuitive distribution of particles. The particle number density is defined as the particle number in each H/12 × H/12 grid cell. The arrangement of grid cells is illustrated in Fig. 6. In the calculation of ρpn, each modified particle is assigned fractionally to its surrounding grid points. The fraction to each point was calculated by the distance between it and the particle shown as follows: ! y −yði; jÞ xp −xði; jÞ p 1− f ði; jÞ ¼ 1− Δy Δx
ð7Þ
where f(i, j) is the fraction to each point, xp and yp are the particle location coordinates, x(i, j) and y(i, j) represent the grid point location, and Δx and Δy are the spaces between two adjacent grid points in the direction of the coordinates. Fig. 7 shows the particle number density distribution for the initial moments of particle motion (flow time = 1.05 s, 1.1 s, 1.15 s, 1.2 s, 1.25 s, and 1.3 s) when h = 0 and St = 1.2. It is obvious from the figure that the upstream cavity is long enough for the particles to be fully distributed. Particles just cross the cavity when the flow time is 1.15 s.
Fig. 7. Particle number density distribution for the initial moments of particle motion when h = 0 and St = 1.2.
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Fig. 8. Streamwise particle velocities at different x positions.
Fig. 9. Particle number density for four inclination angles when h = 0.
Fig. 10. Particle number density for four inclination angles when h = 0.5H.
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Fig. 11. Effect of h on N for four inclination angles.
When the flow time is 1.25 s, particles begin to escape from the domain, that is, the whole domain is full of particles. Thus, particle data for a flow time of 1.3 s is applied in the following analysis. Fig. 8 shows the streamwise particle velocities (up) at five x positions when h = 0 and St = 1.2. Streamwise particle velocities in the interval of [x − 1, x + 1] were regarded as up at x position. Different from gas phase, the distribution of up changes with φ,especially in the shear layer region and the region close to the upper wall of the tube. The change of the streamwise particle velocity will lead to the variation of particle distribution. Figs. 9 and 10 display the distribution of particle number density for four inclination angles. It is noted that the distribution of particles not only changes with φ, but also varies with St and h. So the effects of φ for different St and h are distinct, and details are showing in the following parts. For a Stokes number of 0.12, dominated by drag force, particles show an excellent performance following streamlines. Thus, it is shown in the figures that the particles were uniformly dispersed throughout the cavity, and that the distribution of ρpn was almost independent of φ. For a Stokes number of 1.2, the effect of gravity is increased, and hence the distribution of particles is controlled by the vortex and the gravity force. When h = 0, the main vortex is confined in the cavity, and a large amount of particles are concentrated on its periphery. An increase in φ results in a variation in gy, and thus leads to a decrease in the concentrated particle number. When h reached 0.5H, the limitation
Fig. 12. Influence laws of φ on N for the cases within the shaded area of Fig. 10.
Fig. 13. Variation of negative slopes a1 and a2 with h.
effect of the cavity on the main vortex decreased. Thus, it is obvious that the particle number densities in the cavity are all small, and that the effect of φ on the distribution of ρpn is insignificant. For a Stokes number of 9.1, particle motion is mainly dominated by gravity, the effect of drag force is minimal. When there is no height difference between the leading and aft walls, particles impact on the aft wall and rebound to the bottom of the leading region of the cavity (Fig. 9). With an increase in φ, the particle number density in the cavity decreases. When h = 0.5H, particles flow across the cavity without impacting on the aft wall. Thus, there are almost no particles in the cavity for the discussed inclination angles. Since most practical pipelines are axisymmetric in reality, particle distributions in pipe annular cavities are surmised. When St = 0.12, particles will disperse uniformly in the cavity, and the distribution changes little with h and φ. When St = 1.2, particles will concentrate along the periphery of the main vortex. Owing to the effect of gravity, particle number in the lower half of the annular cavity will be a bit larger than that in the upper half. With the increase of φ, particle number in the lower half of the annular cavity will decrease, while that in the upper half will increase. Meanwhile, the distribution of particles stays the same with the growth of φ. With the increase of h, particle number
Fig. 14. Erosion rate along the aft wall for four inclination angles.
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with φ for the discussed cases. The gradient of the decrease reduces with increasing h. As N is independent with φ when St = 0.12, negative slopes for the other two Stokes numbers were calculated, and curve-fittings were conducted, as shown in Fig. 13. It is indicated that both negative slopes increase exponentially with h. 5.3. Particle erosion
Fig. 15. The maximum erosion rate on the aft wall for four inclination angles.
reduces and the distribution of particles becomes insignificant. When St = 9.1, most particles will distribute in the lower half of the annular cavity when φ is small. With the increases of φ, particle number in the lower half will reduce, and that in the upper half will increases. With the increases of h, particles within the cavity will decrease sharply. As particles in the near wall region are more likely to impact on the wall and lead to erosion, particle numbers in a zone 10 mm from the aft wall were counted (A region in Fig. 1), and are represented by N. Fig. 11 shows the effect of h on N for four inclination angles. For the discussed three St, the particle numbers all reduced with increasing height difference. This decreasing trend is more noticeable for larger St. The figure also indicates that N does not change with φ when St is 0.12. But for St of 1.2 and 9.1, the variation in φ causes a great change in N when h = 0. The effects of φ diminish with increasing h, and vanish at a critical h. Through our simulations, the critical h for St = 1.2 is set at 0.5H, while for St = 9.1 is set at 0.2H. Thus it is obvious that the effective region of φ on N is located at the left bottom of Fig. 11 (shaded region). For discussed h and St within shaded region, the influences of φ on N were extracted, as shown in Fig. 12. It is clear that N decreases linearly
By comparing erosion results of the aft wall and the bottom wall, it is found that the aft wall is more likely to be eroded. When h = 0, φ = 0 and St = 1.2, the maximum erosion rate on the bottom wall is about an order of magnitude smaller than that on the aft wall. Thus, the erosion on the aft wall is analyzed in the following sections. Fig. 14 shows the erosion rate along the aft wall for four inclination angles. It is obvious that the leading edge is eroded the most seriously on the aft wall. Due to the effect of particle motion, it is found that the erosion rate increases with St. Although erosion is not great for smaller St, the cumulative effects can be substantial over tens of years of eroding. It is also demonstrated that the erosion distribution for St of 0.12 is independent of φ. With an increase in St, the effect of φ on erosion distribution becomes more evident. The maximum erosion rate is the major parameter expressing the degree of erosion. Thus, the effects of h on the maximum erosion rates of the aft wall (Em) for four inclination angles are investigated, as presented in Fig. 15. With the influence of particle concentration, the erosion curves are similar to the curves of N (Fig. 12). Em decreased with an increase in h for all the investigated cases, and the reduction is more significant for larger St. Meanwhile, for a St of 0.12, Em is almost invariable with φ, and the effect of φ on Em increases with St. For St of 1.2 and 9.1, the effect of φ reduces, then disappears with increasing h. Affected by particle concentration, the left bottom region in Fig. 15 (shaded region) is also the effective region of φ on the maximum erosion rate. If h varies within the other region (non-shaded region), as mentioned, Em barely changes with φ, the relationship curves being horizontal straight lines. Thus for the shaded region of Fig. 15, the regulation of the maximum erosion rate with an increase in φ is discussed, and displayed in Fig. 16. For the St of 1.2, the same with N-φ curves (Fig. 12), the relationship curves of Em and φ are downward sloping straight lines. This implies that the erosion is almost dominated by the concentration of particles. While for the St of 9.1, different with N-φ curves in
Fig. 16. Regulation of the maximum erosion rate with an increase in φ.
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Fig. 17. Variation of coefficients a3 and a4 with h.
Fig. 12, Em and φ show a quadratic curve relationship. We suspect that the distinction is caused by the impact velocity and the impact angle of particles. For the St of 1.2, the negative slopes (a3) of curves are computed, as shown in Fig. 17. Through curve-fitting, it is found that a3 follows an exponentially increasing with h. The coefficient a4 (representing the feature of the quadratic curve) for the St of 9.1, is also obtained. Curvefitting was also performed for the variations in a4, and a Gaussian curve was obtained.
ERlc et, en Fs FD g H h k L N Ntr St S ṡ u up up1,up2 V vp1,vp2 Wp x y
local erosion ratio coefficients of restitution sharpness factor drag force gravitational acceleration height of leading wall height difference turbulent kinetic energy cavity length particle number in A region number of tracking particles Stokes number of gas-solid flow channel height the sand rate streamwise gas velocity streamwise particle velocity normal component of particle velocity. velocity vector tangential component of particle velocity impact velocity of particle coordinate coordinate
Greek letters α volume fraction φ inclination angle θ impact angle of particle, rad ρ density density of the boundary wall ρw particle number density ρpn
6. Conclusions In this study, the CFD method was used to analyze the effect of the inclination angle on two-phase flow properties and erosion in cavities. The inclination angle ranged from 0 to 90°, the height difference varied from 0 to 0.8H, and the St for the discussed particles were 0.12, 1.2, and 9.1. This study demonstrates that the inclination angle has little effect on gas flow properties (the distribution of streamwise gas velocity (u), the partial derivative of streamwise velocity and the turbulent kinetic energy). But the change of φ will significantly affect particle distribution and the erosion distribution on the aft wall. For the St of 0.12, The particle number (in A region) and the maximum erosion rate on the aft wall barely changes with φ. The effects of φ become more significant with the increase of St, and with the decrease of h. It is also indicated that N decreases linearly with φ for the cases where St = 1.2 and St = 9.1, and the negative slopes of N-φ curves increase exponentially with h. Meanwhile Em decreased linearly with φ when St = 1.2, and decreased in a form of quadratic curve when St = 9.1. The negative slope a3 increases exponentially with h, and the coefficient a4 increases in a Gaussian form with h. Present results provide a reference for the cavity flow designers. And in order to make it a more useful engineering tool, further research will be conducted for 3D structures, multiple particles and multiple surface materials. Notation
Symbols BH dp E Em ER
Brinell hardness particle diameter the erosion rate the maximum erosion rate on the aft wall erosion ratio
Subscripts m maximum n number p particle
Acknowledgment This work has been supported by the National Natural Science Foundation of China (grant no. 51406184), the Public Welfare Technology Application Projects of Zhejiang Province (grant no. 2015C31001) and the Foundation of Zhejiang Sci-Tech University (grant no. 14022005-Y).
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[27] Z. Lin, X. Ruan, Z. Zhu, X. Fu, Numerical study of solid particle erosion in a cavity with different wall heights, Powder Technol. 254 (2014) 150–159. [28] Z. Lin, H. Xu, Y. Wang, Y. Li, B. Cui, X. Ruan, Experimental study of particle erosion in a cavity with a height difference between its walls, Powder Technol. 286 (2015) 378–384. [29] Y.L. Lai, Y.Q. Tai, R. Deng, M.B. Ray, C.H. Wang, Pneumatic transport of granular materials through a 90° bend, Chem. Eng. Sci. 59 (2013) 4637–4651. [30] K.F. Yu, K.S. Lau, C.K. Chan, Numerical simulation of gas-particle flow in a single-side backward-facing step flow, J. Comput. Appl. Math. 163 (2004) 319–331. [31] C.B. Solnordal, C.Y. Wong, A. Zamberi, M. Jadid, Z. Johar, Determination of erosion rate characteristic for particles with size distributions in the low Stokes number range, Wear 305 (2013) 205–215. [32] J.R. Fessler, J.K. Eaton, Particle response in a planar sudden expansion flow, Exp. Thermal Fluid Sci. 15 (1997) 413–423. [33] H.A. Jakobsen, Chemical Reactor Modeling: Multiphase Reactive Flows, Second ed Springer, Switzerland, 2008. [34] G. Grant, W. Tabakoff, Erosion prediction in turbomachinery resulting from environmental solid particles, J. Aircr. 12 (1975) 471–478. [35] K.R. Ahlert, Effects of Particle Impingement Angle and Surface Wetting on Solid Particle Erosion of AISI 1018 Steel, University of Tulsa, 1994. [36] B.S. McLaury, Predicting Solid Particle Erosion Resulting from Turbulent Fluctuations in Oilfield Geometries, University of Tulsa, 1996. [37] J. Wang, S.A. Shirazi, A CFD based correlation for erosion factor for long-radius elbows and bends, J Energ Resour-T ASME 125 (2003) 26–34. [38] X. Chen, B.S. McLaury, S.A. Shirazi, A comprehensive procedure to estimate erosion in elbows for gas/liquid/sand multiphase flow, J Energ Resour-T ASME 128 (2006) 70–78. [39] S.M. El-Behery, M.H. Hamed, M. El-Kadi, K. Ibrahim, Numerical simulation and CFDbased correlation of erosion threshold gas velocity in pipe bends, CFD Lett. 2 (2010) 39–53.
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
Prediction of particle distribution and particle impact erosion in inclined cavities Zhe Lin a,b,⁎, Yifan Zhang a, Yi Li a, Xiaojun Li a, Zuchao Zhu a a b
Key Laboratory of Fluid Transmission Technology of Zhejiang Province, Zhejiang Sci-Tech University, 5 Second Avenue, Xiasha Higher Education Zone, Hangzhou 310018, China Zhejiang Institute of Mechanical & Electrical Engineering Co., Ltd., Hangzhou 310011, China
a r t i c l e
i n f o
Article history: Received 7 February 2016 Received in revised form 18 September 2016 Accepted 17 October 2016 Available online 19 October 2016 Keywords: Cavity Inclination angle Solid particle erosion Computational methods
a b s t r a c t The cavity is one of the basic structures in pipe systems, but it is vulnerable in erosive environments. Cavities can occur in various places throughout industrial processes, each making a difference in the distribution of erosion. The aim of this study is to investigate gas (air)-solid flow properties and erosion characteristics in inclined cavities. A two-way URANS (SST)-DPM method was adopted. The simulations were carried out for three Stokes numbers (St), eight height differences (h) from 0 to 0.8H, and ten inclination angles from 0° to 90°. Results indicated that gas flow properties change little with inclination angle (φ). However, a variation in φ resulted in a change in the distribution of the particles and erosion on the aft wall, especially when St were 1.2 and 9.1. The particle number (N) in the zone 10 mm from the aft wall decreases linearly with φ for both previously mentioned St, and the slopes of N-φ curves increase exponentially with h. The maximum erosion rate (Em) on the aft wall also decreases linearly with φ for the St of 1.2, and the negative slope of Em-φ curve follows an exponentially increasing with h. But Em decreases quadratically with φ when St is 9.1, and the function coefficient appears a Gaussian relationship with h. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Solid particle erosion is inevitable in pipeline transportation with the working fluids containing solid particles, such as pneumatic conveying [1–3]. It is also a key factor affecting the lifetime of piping components and costs industries millions of dollars each year [4]. In past decades, to obtain erosion characteristics, a large amount of investigation has been carried out on piping components such as pipes, elbows, sudden expansion, sudden contraction, and valves. Shirazi et al. [5] developed a simple semi-empirical procedure for the estimation of erosion in piping components, and demonstrated the procedure for a tee and an elbow. Gabriel et al. [6] and Carlos et al. [7] revealed the effects of sand particle concentrations on the erosion of an elbow by simulations. Hanson et al. [8], Niu et al. [9], McLaury et al. [10,11], Akilli et al. [12], Yong Quek et al. [13] and Brown [14] also discussed the erosion in elbows by using simulation and experimental methods. Lee et al. [15] used the Eulerian approach to predict erosion in a boiler tube. Habib et al. [16–18] used the CFD method to investigate the effect of fluid flow and geometry on the erosion of a pipe contraction. Nemitallah et al. [19] indicated the effects of flow velocity, particle size, and pipe material on the downstream erosion of a sharp-edged orifice. Shabgard et al. [20] investigated the microstructural erosion ⁎ Corresponding author at: Key Laboratory of Fluid Transmission Technology of Zhejiang Province, Zhejiang Sci-Tech University, China. E-mail address: [email protected] (Z. Lin).
http://dx.doi.org/10.1016/j.powtec.2016.10.035 0032-5910/© 2016 Elsevier B.V. All rights reserved.
features and material removal mechanisms of AISI H13 core boxes using experimental methods. Nøkleberg and Søntvedt [21] developed a model, which verified with experimental results, to estimate the erosion and lifetime of chokes valves. Forder et al. [22] presented a computational fluid dynamics erosion model and predicted erosion rates within oilfield control valves. McLaury et al. [23] performed a computational study to examine the erosion in a choke geometry; they demonstrated that abrupt changes in a fitting can result in regions of high erosion. Wallace et al. [24] used a Eulerian–Lagrangian model to examine erosion in two choke valves. Cavities (Fig. 1) are one of the basic structures in pipe systems (such as in double flat gate valves). But only a few studies exist concerning particle erosion in cavities, including our own previous studies. Postlethwaite and Nesic [25] tested the erosion in an ideal cavity (where the height of the leading wall is equal to that of the aft wall) using silica sand. Wong et al. [26] investigated the erosion in a vertical ideal annular cavity (i.e., where the angle between the gravity force and the tube direction, φ, is 0°) using CFD and experimental methods. We also conducted both experimental and numerical studies investigating the characteristics of erosion in horizontal (φ = 90°) cavities. The effects of the height difference between the leading wall and aft wall on particle distribution and erosion have been analyzed [27,28]. From the above review, it is believed that previous studies have only focused on horizontal and vertical cavities. However, cavities have various orientations in process industries. Therefore, this study investigates gas-solid flow properties and erosion characteristics in inclined cavities
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in this region do not impact the distribution of particles in the area away from the bottom wall, and also have no effect on the erosion of the aft wall. For the region near the aft wall, the largest local maximum volume fraction is about 0.1% for the discussed cases. According to Hugo [33], two-way coupling is considered a reasonable approach for αp ≤ 0.1%. Therefore, particle collisions were neglected in present simulations, and two-way Eulerian-Lagrangian simulation method was used to predict the erosion of the aft wall within the cavity structure. There are three main models in the present simulation as follows: the continuous phase model used for the prediction of the carrier fluid flow-field; the particle tracking model used for the calculation of particle trajectories; and the erosion model used for the determination of degree of erosion. This study is the follow-up research of our previous work [27]. The simulation method and the simulation procedure of this study are the same as that of our previous work. Thus, the simulation method and the procedure are described briefly in this study, further details were given in our previous paper. Fig. 1. Schematic structure of the computational domain.
3.1. The continuous phase model
(Fig. 1). The Eulerian-Lagrangian simulation method is applied. The effects of the inclination angle (φ) on gas flow, particle distribution, and erosion of the aft wall are discussed.
The dynamics of incompressible flow were solved using the incompressible unsteady Reynolds-Averaged Navier-Stokes equation. The turbulence is calculated by the shear-stress transport (SST) model. 3.2. The particle tracking model
2. Problem statement The problem considered is that of solid particle erosion in an inclined cavity. In accordance with our previous study [27], a 2D computational domain is used in this paper, as shown in Fig. 1. The flow domain consists of a cavity structure and two extension tubes. The heights of the upstream tube (S) and the leading wall (H) are 40 mm and 26.7 mm, respectively. According to the upstream length function of pipe turbulent flow, and referred to the two phase studies [29,30], the tube length upstream the cavity (Lup) was set as 32S. Meanwhile, the velocity profiles at the inlet of the cavity for Lup of 32S and 150S were also compared, and the results indicated that the upstream length of 32S is reasonable for present simulations. The tube length downstream of the cavity was set as 35H. The length of the cavity (L) was three times of H. The top point of the leading wall was set as the origin of the coordinates. Discussed height differences (h) were 0, 0.1H, 0.2H, 0.3H, 0.4H, 0.5H, 0.6H, and 0.8H. The inclination angle (φ) varied from 0° to 90°. Air at 25 °C is regarded as the carrier fluid. The working pressure is assumed as 1.01 × 105 Pa. The average gas velocity at the inlet was 9.3 m/s (i.e., the centerline velocity was 10.5 m/s when the flow developed fully). As the Mach number was far b 0.3, the carrier fluid was set as incompressible. The density and the dynamic viscosity of air were set as 1.225 kg/m3 and 1.79 × 10−5 Pa s, respectively. Stokes numbers (St) directly represent the following behaviors of particles with carrier fluid. The change of St will affect the distribution of particles and particle erosions [27,28,31]. Thus, to investigate the effect of St, diameters of 15, 50, and 150 μm were employed. According to Fessler and Eaton [32], the corresponding Stokes numbers (St) are 0.12, 1.2, and 9.1 (details are presented in our previous study [27]). The density of particles is 2500 kg ⁄ m3. The investigated particle mass loading ratio is 0.2, that the volume fraction (αp) of loading particles is 0.0098% (i.e. 98 ppm by volume). The pipe material was set as carbon steel, and the Brinell hardness number was 140. 3. Simulation modeling For the investigated cases, there are two high particle density regions inside the cavity: the bottom region of the cavity and the region near the aft wall. For the bottom region, particle velocities are extremely small. So whether consider particle collisions or not, motions of particles
Particle trajectories were calculated using Newton's second law. As the Reynolds number is large in the present study, the Magnus lift force was not considered. The high value of ρp/ρ implies that the pressure gradient force and the virtual mass force were small and were therefore neglected. The Basset history force was ignored due to the low particle acceleration in simulations. By the magnitude analysis, it is found that the Saffman lift force was two orders of magnitude smaller than the drag force. Thus, the Saffman lift force on particles was also neglected. Under these assumptions, only drag and gravitation forces were used in the present calculations, shown as Eq. (1). The Random Walk Model (RWM) was also adopted for the calculation of the turbulent dispersion of particles. dV p ¼ FD þ dt
! ρ g 1− ρp
ð1Þ
where the first term on the right-hand side is the drag force, and the second term is the gravity force. Particle-wall interaction plays an important role in the calculation of particle trajectories. Hence, the coefficients of restitution (et, en) presented by Grant [34] were used to describe the collision and rebound performance between particle and wall. en ¼ 0:993−1:76θ þ 1:56θ2 −0:49θ3
ð2Þ
et ¼ 0:988−1:66θ þ 2:11θ2 −0:67θ3
ð3Þ
where θ is the impact angle of the particle. Table 1 Erosion model empirical constants for carbon steel (dry surface) and aluminum [36]. Constant
Carbon steel
Aluminum
C α a b w X Y Z
1.22 × BH−0.59 15 deg −3.34 × 10−4 1.79 × 10−4 1 1.239 × 10−5 −1.192 × 10−5 2.167 × 10−5
1.865 × 10−6 10 deg −34.79 12.3 5.205 0.147 −0.745 1
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Fig. 2. The nephogram of the streamwise velocity and the streamline of the gas phase for h = 0.
The erosion rate (mm/h), E, is calculated according to the equation of Shirazi et al. [5].
3.3. The erosion model The widely used semi-empirical single particle erosion model proposed by Ahlert [35] and McLaury [36] was adopted for the calculation of the erosion ratio. This model is relevant for the erosion study of piping components (including cavities) [26,37–39]. ER ¼ C F s W p n f ðθÞ f ðθÞ ¼
ð4Þ
E ¼ 3:6 106
s_ ER ρw Ntr A lc
ð6Þ
where ṡ denotes the sand rate (kg/s), ρw is the density of the boundary wall (kg/m3), Ntr is the number of tracking particles, A is the impingement area (m2) and ERlc is the local erosion ratio (kg/kg). 4. Simulation procedure and validation
2
aθ þ bθ Xcos2 ðθÞ sinðwθÞ þ Ysin2 ðθÞ þ Z
0≤θbα α ≤θb90 °
ð5Þ
where ER is the erosion ratio, defined as the mass eroded of the wall surface divided by the mass of the particles impacting the wall (kg/kg). Fs is the particle sharpness factor; Fs = 1 represents sharp particles, 0.53 semi-rounded particles, and 0.2 rounded particles. W is the particle impingement velocity (m/s). The value of velocity exponent n is 1.73. The empirical constants for carbon steel and aluminum are given in Table 1 [36].
The commercial software FLUENT (Ansys, Inc., Canonsburg, PA, USA) was used for numerical calculations. The velocity-inlet boundary condition (9.3 m/s) and the pressure-outlet boundary condition (1.01 × 105 Pa) were applied for the simulations. The second-order central difference scheme was applied for pressure gradient, convection terms, and divergence terms. The automated tracking scheme (switches automatically between the implicit scheme and the trapezoidal scheme) was employed for the calculation of particle motion. The gas phase and the particle phase have a same transient time-step of 0.0005 s.
Fig. 3. Gas streamwise velocities at different x positions.
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Fig. 4. Partial derivatives of streamwise velocity at different x positions.
Firstly, single gas flow over the cavity was calculated using the SIMPLEC algorithm, and this procedure ceased at a flow time of 1 s. We then input 40 uniformly distributed parcels per time-step into the flow field. For each time-step, the coupling calculation between the two phases continued until there were no more changes in particle source and gas-phase quantities between the two iterations. Meanwhile, particle impact erosions on the cavity walls for each time-step were recorded, and then erosions during 1.5–2.5 s were calculated. Structural mesh was used in the mesh modeling, and the boundary layer was refined to meet the requirements of the simulation model. A grid-independence test was performed in our previous study using 5 grid numbers (30,000, 100,000, 200,000, 360,000, and 660,000.) [27]. The result showed that more there was little change in simulation
results when the grid number is larger than 200,000. Thus, a grid number of 200,000 was used for the present calculations. The validation of our computational procedure comprises the following two parts: two-phase flow validation and erosion validation. To verify the accuracy of two-phase flow properties, the simulated streamwise mean velocities and streamwise fluctuating velocities of a vertically positioned backward-facing step were compared with the experimental data obtained by Fessler and Eaton [32]. Good agreements were achieved between our simulation results and the experimental data. For the validation of erosion, the calculation results of a vertically positioned cavity were compared with the experimental erosion rates (the depth of the eroded material per time interval) of Wong [26]. The
Fig. 5. Turbulent kinetic energies at different x positions.
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Fig. 6. The arrangement of grid cells for the calculation of particle number density.
experiments were carried out with the mass flow rate of 0.03 kg/s, and the results were recorded between different particle loadings, i.e. different time period. The 0–25 kg data set could be affected by the surface inclusions (such as the oxide layer), and the 0–300 kg data set could be influenced by the erosion morphology. Thus the 25–50 kg data set is more suitable for erosion validation. The comparison shows that our calculation erosion rates agree well with the 25–50 kg experimental data. And the average error of predictions between y/H = −0.3 and 0 is lower than 15%. Details of validations are given in our previous study [27]. 5. Results and discussion Gas-solid flow properties and erosion characteristics in inclined cavities were examined using a two-way Eulerian–Lagrangian simulation method. The effects of inclination angle on the flow of the gas phase, the motion of particles and the particle erosion were investigated. 5.1. Gas phase The flow of the gas can be reflected by the distribution of streamwise gas velocity (u), partial derivative of streamwise velocity (∂u/∂y) and turbulent kinetic energy (k). Thus, these flow characteristics for the discussed cases are extracted and compared. Comparisons show that the inclination angle has a negligible effect on the flow states of the
gas phase. This is caused by the small mass loading ratio (0.2) used in this study. Therefore, the flow characteristics for h = 0 and St = 1.2 are selected and displayed in the following section, to show the effect of φ quantitatively. When the gas passes through the ideal cavity (h = 0), a large main vortex forms in the aft part of the cavity by the shear layer (generated on the top area of the cavity), as shown in Fig. 2. And another opposite rotating vortex (secondary vortex) arose in the leading part. Meanwhile, there are two small vortexes existed at the bottom corners of the cavity. To explore the influence of φ, we extracted streamwise velocities at x = 0.05L, 0.2L, 0.5L, 0.8L and 0.95L, as plotted in Fig. 3. It is noted that the velocity curves for four inclination angles are coincident. Fig. 4 shows the partial derivatives of streamwise velocity (∂u/∂y) at previously mentioned five positions. It is clear that the shear layer region has a large ∂u/∂y, and the value decreases with the increase of x. Meanwhile, determined by the vortexes, the values of ∂u/∂y in the bottom areas of the cavity change greatly with x. By comparison, the inclination angle has little effect on ∂u/∂y. The distribution of turbulent kinetic energy presents the turbulence of gas flow. In the presence of shear layer, the turbulent kinetic energies in the top area of the cavity are larger than other positions (Fig. 5). Meanwhile, it is obvious that the distribution of k is independent on inclination angle. 5.2. Particle motion The concept of particle number density (ρpn) was employed to get a more intuitive distribution of particles. The particle number density is defined as the particle number in each H/12 × H/12 grid cell. The arrangement of grid cells is illustrated in Fig. 6. In the calculation of ρpn, each modified particle is assigned fractionally to its surrounding grid points. The fraction to each point was calculated by the distance between it and the particle shown as follows: ! y −yði; jÞ xp −xði; jÞ p 1− f ði; jÞ ¼ 1− Δy Δx
ð7Þ
where f(i, j) is the fraction to each point, xp and yp are the particle location coordinates, x(i, j) and y(i, j) represent the grid point location, and Δx and Δy are the spaces between two adjacent grid points in the direction of the coordinates. Fig. 7 shows the particle number density distribution for the initial moments of particle motion (flow time = 1.05 s, 1.1 s, 1.15 s, 1.2 s, 1.25 s, and 1.3 s) when h = 0 and St = 1.2. It is obvious from the figure that the upstream cavity is long enough for the particles to be fully distributed. Particles just cross the cavity when the flow time is 1.15 s.
Fig. 7. Particle number density distribution for the initial moments of particle motion when h = 0 and St = 1.2.
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Fig. 8. Streamwise particle velocities at different x positions.
Fig. 9. Particle number density for four inclination angles when h = 0.
Fig. 10. Particle number density for four inclination angles when h = 0.5H.
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Fig. 11. Effect of h on N for four inclination angles.
When the flow time is 1.25 s, particles begin to escape from the domain, that is, the whole domain is full of particles. Thus, particle data for a flow time of 1.3 s is applied in the following analysis. Fig. 8 shows the streamwise particle velocities (up) at five x positions when h = 0 and St = 1.2. Streamwise particle velocities in the interval of [x − 1, x + 1] were regarded as up at x position. Different from gas phase, the distribution of up changes with φ,especially in the shear layer region and the region close to the upper wall of the tube. The change of the streamwise particle velocity will lead to the variation of particle distribution. Figs. 9 and 10 display the distribution of particle number density for four inclination angles. It is noted that the distribution of particles not only changes with φ, but also varies with St and h. So the effects of φ for different St and h are distinct, and details are showing in the following parts. For a Stokes number of 0.12, dominated by drag force, particles show an excellent performance following streamlines. Thus, it is shown in the figures that the particles were uniformly dispersed throughout the cavity, and that the distribution of ρpn was almost independent of φ. For a Stokes number of 1.2, the effect of gravity is increased, and hence the distribution of particles is controlled by the vortex and the gravity force. When h = 0, the main vortex is confined in the cavity, and a large amount of particles are concentrated on its periphery. An increase in φ results in a variation in gy, and thus leads to a decrease in the concentrated particle number. When h reached 0.5H, the limitation
Fig. 12. Influence laws of φ on N for the cases within the shaded area of Fig. 10.
Fig. 13. Variation of negative slopes a1 and a2 with h.
effect of the cavity on the main vortex decreased. Thus, it is obvious that the particle number densities in the cavity are all small, and that the effect of φ on the distribution of ρpn is insignificant. For a Stokes number of 9.1, particle motion is mainly dominated by gravity, the effect of drag force is minimal. When there is no height difference between the leading and aft walls, particles impact on the aft wall and rebound to the bottom of the leading region of the cavity (Fig. 9). With an increase in φ, the particle number density in the cavity decreases. When h = 0.5H, particles flow across the cavity without impacting on the aft wall. Thus, there are almost no particles in the cavity for the discussed inclination angles. Since most practical pipelines are axisymmetric in reality, particle distributions in pipe annular cavities are surmised. When St = 0.12, particles will disperse uniformly in the cavity, and the distribution changes little with h and φ. When St = 1.2, particles will concentrate along the periphery of the main vortex. Owing to the effect of gravity, particle number in the lower half of the annular cavity will be a bit larger than that in the upper half. With the increase of φ, particle number in the lower half of the annular cavity will decrease, while that in the upper half will increase. Meanwhile, the distribution of particles stays the same with the growth of φ. With the increase of h, particle number
Fig. 14. Erosion rate along the aft wall for four inclination angles.
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with φ for the discussed cases. The gradient of the decrease reduces with increasing h. As N is independent with φ when St = 0.12, negative slopes for the other two Stokes numbers were calculated, and curve-fittings were conducted, as shown in Fig. 13. It is indicated that both negative slopes increase exponentially with h. 5.3. Particle erosion
Fig. 15. The maximum erosion rate on the aft wall for four inclination angles.
reduces and the distribution of particles becomes insignificant. When St = 9.1, most particles will distribute in the lower half of the annular cavity when φ is small. With the increases of φ, particle number in the lower half will reduce, and that in the upper half will increases. With the increases of h, particles within the cavity will decrease sharply. As particles in the near wall region are more likely to impact on the wall and lead to erosion, particle numbers in a zone 10 mm from the aft wall were counted (A region in Fig. 1), and are represented by N. Fig. 11 shows the effect of h on N for four inclination angles. For the discussed three St, the particle numbers all reduced with increasing height difference. This decreasing trend is more noticeable for larger St. The figure also indicates that N does not change with φ when St is 0.12. But for St of 1.2 and 9.1, the variation in φ causes a great change in N when h = 0. The effects of φ diminish with increasing h, and vanish at a critical h. Through our simulations, the critical h for St = 1.2 is set at 0.5H, while for St = 9.1 is set at 0.2H. Thus it is obvious that the effective region of φ on N is located at the left bottom of Fig. 11 (shaded region). For discussed h and St within shaded region, the influences of φ on N were extracted, as shown in Fig. 12. It is clear that N decreases linearly
By comparing erosion results of the aft wall and the bottom wall, it is found that the aft wall is more likely to be eroded. When h = 0, φ = 0 and St = 1.2, the maximum erosion rate on the bottom wall is about an order of magnitude smaller than that on the aft wall. Thus, the erosion on the aft wall is analyzed in the following sections. Fig. 14 shows the erosion rate along the aft wall for four inclination angles. It is obvious that the leading edge is eroded the most seriously on the aft wall. Due to the effect of particle motion, it is found that the erosion rate increases with St. Although erosion is not great for smaller St, the cumulative effects can be substantial over tens of years of eroding. It is also demonstrated that the erosion distribution for St of 0.12 is independent of φ. With an increase in St, the effect of φ on erosion distribution becomes more evident. The maximum erosion rate is the major parameter expressing the degree of erosion. Thus, the effects of h on the maximum erosion rates of the aft wall (Em) for four inclination angles are investigated, as presented in Fig. 15. With the influence of particle concentration, the erosion curves are similar to the curves of N (Fig. 12). Em decreased with an increase in h for all the investigated cases, and the reduction is more significant for larger St. Meanwhile, for a St of 0.12, Em is almost invariable with φ, and the effect of φ on Em increases with St. For St of 1.2 and 9.1, the effect of φ reduces, then disappears with increasing h. Affected by particle concentration, the left bottom region in Fig. 15 (shaded region) is also the effective region of φ on the maximum erosion rate. If h varies within the other region (non-shaded region), as mentioned, Em barely changes with φ, the relationship curves being horizontal straight lines. Thus for the shaded region of Fig. 15, the regulation of the maximum erosion rate with an increase in φ is discussed, and displayed in Fig. 16. For the St of 1.2, the same with N-φ curves (Fig. 12), the relationship curves of Em and φ are downward sloping straight lines. This implies that the erosion is almost dominated by the concentration of particles. While for the St of 9.1, different with N-φ curves in
Fig. 16. Regulation of the maximum erosion rate with an increase in φ.
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Fig. 17. Variation of coefficients a3 and a4 with h.
Fig. 12, Em and φ show a quadratic curve relationship. We suspect that the distinction is caused by the impact velocity and the impact angle of particles. For the St of 1.2, the negative slopes (a3) of curves are computed, as shown in Fig. 17. Through curve-fitting, it is found that a3 follows an exponentially increasing with h. The coefficient a4 (representing the feature of the quadratic curve) for the St of 9.1, is also obtained. Curvefitting was also performed for the variations in a4, and a Gaussian curve was obtained.
ERlc et, en Fs FD g H h k L N Ntr St S ṡ u up up1,up2 V vp1,vp2 Wp x y
local erosion ratio coefficients of restitution sharpness factor drag force gravitational acceleration height of leading wall height difference turbulent kinetic energy cavity length particle number in A region number of tracking particles Stokes number of gas-solid flow channel height the sand rate streamwise gas velocity streamwise particle velocity normal component of particle velocity. velocity vector tangential component of particle velocity impact velocity of particle coordinate coordinate
Greek letters α volume fraction φ inclination angle θ impact angle of particle, rad ρ density density of the boundary wall ρw particle number density ρpn
6. Conclusions In this study, the CFD method was used to analyze the effect of the inclination angle on two-phase flow properties and erosion in cavities. The inclination angle ranged from 0 to 90°, the height difference varied from 0 to 0.8H, and the St for the discussed particles were 0.12, 1.2, and 9.1. This study demonstrates that the inclination angle has little effect on gas flow properties (the distribution of streamwise gas velocity (u), the partial derivative of streamwise velocity and the turbulent kinetic energy). But the change of φ will significantly affect particle distribution and the erosion distribution on the aft wall. For the St of 0.12, The particle number (in A region) and the maximum erosion rate on the aft wall barely changes with φ. The effects of φ become more significant with the increase of St, and with the decrease of h. It is also indicated that N decreases linearly with φ for the cases where St = 1.2 and St = 9.1, and the negative slopes of N-φ curves increase exponentially with h. Meanwhile Em decreased linearly with φ when St = 1.2, and decreased in a form of quadratic curve when St = 9.1. The negative slope a3 increases exponentially with h, and the coefficient a4 increases in a Gaussian form with h. Present results provide a reference for the cavity flow designers. And in order to make it a more useful engineering tool, further research will be conducted for 3D structures, multiple particles and multiple surface materials. Notation
Symbols BH dp E Em ER
Brinell hardness particle diameter the erosion rate the maximum erosion rate on the aft wall erosion ratio
Subscripts m maximum n number p particle
Acknowledgment This work has been supported by the National Natural Science Foundation of China (grant no. 51406184), the Public Welfare Technology Application Projects of Zhejiang Province (grant no. 2015C31001) and the Foundation of Zhejiang Sci-Tech University (grant no. 14022005-Y).
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