Numerical prediction of particle erosion of pipe bends

APT 2152

No. of Pages 18, Model 5G

26 November 2018 Advanced Powder Technology xxx (xxxx) xxx 1

Contents lists available at ScienceDirect

Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt

2

Original Research Paper

5 4 6

Numerical prediction of particle erosion of pipe bends

7 8 9 10 12 11 13 1 7 5 2 16 17 18 19 20 21 22 23 24 25 26

S. Laín a, M. Sommerfeld a,b a b

PAI+, Energetics and Mechanics Department, Universidad Autónoma de Occidente, Cali, Colombia MPS (Multiphase Flow Systems), Faculty of Process and Systems Engineering, Otto-von-Guericke University, Halle (Saale), Germany

a r t i c l e

i n f o

Article history: Received 8 March 2018 Received in revised form 27 October 2018 Accepted 15 November 2018 Available online xxxx Keywords: Pneumatic conveying Erosion Four-way coupling Wall roughness

a b s t r a c t In the present study the Euler/Lagrange approach in combination with a proper turbulence model and full two-way coupling is applied for erosion estimation due to particle conveying along a horizontal to vertical pipe bend. Particle tracking considers both particle translational and rotational motion and all relevant forces such as drag, gravity/buoyancy and transverse lift due to shear and particle rotation were accounted for Laín and Sommerfeld (2012). Moreover, models for turbulent transport of the particles, collisions with rough walls and inter-particle collisions using a stochastic approach are considered Sommerfeld and Laín (2009). In this work, the different transport effects on spherical solid particle erosion in a pipe bend of a pneumatic conveying system are analysed. For describing the combined effect of cutting and deformation erosion the model of Oka et al. (2005) is used. Erosion depth was calculated for two- and four-way coupling and for mono-sized spherical glass beads as well as a size distribution of particles with the same number mean diameter (i.e. 40 lm). Additionally, particle mass loading was varied in the range from 0.3 to 1.0. The erosion model was validated on the basis of experiments by Mazumder et al. (2008) for a narrow vertical to horizontal pipe system with high conveying velocity. Then a 150 mm pipe system with 5 m horizontal pipe, pipe bend and 5 m vertical pipe with a bulk velocity of 27 m/s was considered for further analysis. As a result inter-particle collisions reduce erosion although the wall collision frequency is enhanced Sommerfeld and Laín (2015); additionally, considering a particle size distribution with the same number mean diameter as mono-sized particles yields much higher erosion depth. Finally, when particle mass loading is increased, bend erosion is reduced due to modifications of particle impact velocity and angle, although wall collision frequency grows. Ó 2018 Published by Elsevier B.V. on behalf of The Society of Powder Technology Japan. All rights reserved.

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

1. Introduction

53

Transport of particulate material in confined environments is widely spread in a variety of chemical and industrial processes. Examples of industries which employ conveying of bulk materials include agriculture, mining, chemical, paint manufacture, metal refining or pharmaceutical. The main advantages of this transportation technique are its high flexibility, reliability and easy maintenance together with a low material degradation when the appropriate system is employed. However, as a consequence of the solids conveying along the pipeline associated with numerous wall impacts, particulate erosion takes place on the walls of the transport line. Following El Togby and Elbestawi [6] such erosion of materials caused by the impact of hard particles is one of the forms of material degradation classified as wear. There exist several definitions of erosion.

54 55 56 57 58 59 60 61 62 63 64 65 66

E-mail addresses: [email protected] (S. Laín), [email protected] (M. Sommerfeld)

According to Bitter [7] it is defined as ‘‘material damage caused by the attack of particles entrained in a fluid system impacting the surface at high speed” while Hutchings and Winter [8] defines it as: ‘‘erosion is an abrasive wear process in which the repeated impact of small particles entrained in moving fluid against a surface results in the removal of material from that surface”. In oil and gas industry, for instance, the problems derived from wear caused by particle impacts are especially severe, as they adversely influence not only the transport process efficiency but also compromise safety [9]. Sand particles removed when extracting oil threaten the integrity of the transport and process components such as pipes, valves and separation devices: a failure, due to erosion, in such elements might derive for example in oil spill which can generate an environmental impact as well as a considerable liability of the company [10]. Therefore, it is of paramount importance to develop strategies aimed to predict erosion rate depending on operating conditions [11]. It would not only allow the service life estimation but also to preview the geometry locations were critical erosion is likely to occur. Nevertheless, this is

https://doi.org/10.1016/j.apt.2018.11.014 0921-8831/Ó 2018 Published by Elsevier B.V. on behalf of The Society of Powder Technology Japan. All rights reserved.

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85

APT 2152

No. of Pages 18, Model 5G

26 November 2018 2 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151

S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

not an easy task as particulate erosion depends on numerous factors some of which contribute in a synergistic manner to intensify wall material degradation. Nowadays, it is accepted that solid particle erosion acts through two main mechanisms depending on the ductility of the surface. For ductile materials the main mechanisms is micro-cutting together with work hardening of the surface [12,13]. For brittle materials, it is accepted that erosion is due to crack formation [14]. A more detailed perspective of both kinds of erosion mechanisms is found in Parsi et al. [15], which also provide an overview of the most relevant parameters influencing erosion. These authors include particle physical properties (shape, size and material), fluid properties, target wall properties, particle impact speed and angle and inter-particle collisions. Remarkably, angular particle shapes can result in erosion four times larger than that of round particles [16] and the impact angle resulting in maximum erosion varies with particle angularity. Larger particles causes higher erosion ratio (defined as mass of eroded material over the mass of impacting particles) than smaller ones, because they have larger kinetic energy; however, the erosion ratio is found to be nearly independent of particle size for particles larger than 100 lm. Though, particle density, shape, and hardness also affect erosiveness, but in general larger sand particles cause more erosion damage for similar impact speed, and shape, density, and hardness. Smaller sand particles are also more affected by turbulence. The exchange of momentum between fluid and particle is more efficient for smaller particles, so they respond to fluctuation in the flow more easily. For particles with similar size and shape, higher density and hardness usually cause higher erosion rates (depending on hardness of the target material). Higher particle density increases the kinetic energy and impact force and increases the erosion rate. Larger hardness of the particles (up about 700 Hv, Vickers hardness) also increases the erosion rate. Carrier fluid may also have a large influence on erosion ratio because it affects impact velocities of particles on wall target material. For instance, the slip velocity of solids in air is much larger than in water. Also, fluid properties directly affect the local particle concentration, which can be high due to the flow pattern affecting both, erosion magnitude and pattern. During long time, it was believed that materials with lower hardness were more prompt to erosion than those with higher hardness. However, Levy and Hickey [17] found in some cases the opposite trend. These authors suggest using toughness of the target material as a better indicator of erosion performance than hardness. As a result and after numerous experiments, nowadays there is no definite correlation between target wall material and particle erosion rate. The particle impact velocity strongly determines the erosion rate. The relation between them follows a potential relation, with a constant exponent n, whose value can vary between 0.3 and 4.5 depending on the investigator, although the theoretical value is 2.0. However, it has been suggested that such an exponent is not constant but depends on the hardness of the eroded material [3,18]. Typical values for n are between 1.6 and 2.6. Another important influence on erosion rate is particle impact angle, coupled however with the type of wall material used. For brittle materials the main erosion mechanism is the formation of radial and lateral cracks when particles hit the surface, so the maximum erosion happens for near normal impacts corresponding to angles close to 90°. On the other hand, in the case of ductile surfaces the cutting action and subsequent crater formation is predominant, so the maximum erosion values are obtained for more shallow angles. However, most material surfaces have characteristics of both, ductile and brittle materials, consequently a variety of angle functions have been proposed in the literature. As a result,

most of the proposed angle functions are empirical and are only valid for limited conditions. The effect of particle-particle interactions on erosion has been observed by several researchers since the 70s, being the majority of them empirical studies. Uuemis and Kleis [19] observed a reduction in the specific erosion rate (mass of material removed from the surface per unit mass of particles impacting the wall), with increasing particle concentration (or flux), even though the absolute rate of erosion (total eroded mass) increased with increasing concentration. In the case of pneumatic gas-solid conveyors, Mills [20] showed that the total mass loss from the entire bend, after conveying a fixed quantity of material, decreased when the suspension density (i.e., concentration) of particles in the bend increased. Several researches (e.g. [21]) have proposed different mechanisms which explain the relation between erosion rate and particle concentration. It has been suggested that at high concentrations, when particles rebound from the wall, they hit particles that move toward the wall and slow them down. This phenomenon is named ‘‘shielding” or ‘‘cushioning” [22]. Therefore, not always higher particle concentrations imply higher erosion rate, but it is strongly dependent on the fluid and geometry conditions. Such phenomena have also been found recently in numerical simulations of particulate flow along elbows [23–25]. A powerful methodology to estimate the erosion caused by particles in internal components of process systems is to couple twophase Eulerian-Lagrangian CFD computations with an appropriate solid particle erosion parametrization. In that way, the required input parameters needed by the erosion model, namely particle speed and impact angle at contact, are readily available from the Lagrangian tracking of the CFD simulation. This approach is called CFD based erosion modelling and consists in the following steps: flow computation, particle tracking and applying erosion equations [10]. Ref. [15] performed a review of the studies that employ such a methodology. The main advantage of the CFD-based erosion analysis is that the different factors influencing erosion can be studied separately or in appropriate combinations of them to find out the areas more likely to experience strong material loss, as well as to predict the maximum erosion rates even in geometries where setting an experimental rig is difficult. In performing CFD based erosion analysis it is customary to adopt some simplifications such as neglecting particle size distribution and inter-particle collisions, e.g. [26–29]. Considering inter-particle collisions is numerically very expensive or requires suitable models (see e.g. [30,31]). Moreover, in many studies particle rotation and transverse lift forces due to shear and rotation are not considered [10,26,28]. Especially, in confined flows which are wall-collision dominated this is not justified (see for example [2,30,32]). For particles larger than about 50 lm in a gas flow transverse lift forces should be considered [30,33], which will eventually also influence the wall collision frequency and hence erosion [32]. It was also shown in the past that inter-particle collisions have a drastic effect on particle behavior in different pipe elements and thereby also on wall collision frequency [1,34]. The assumption mostly made, that the over-all particle volume fraction is very small and therefore, two-way coupling and inter-particle collisions may be neglected is often simply wrong [27,35]. Considering glass powder (qp = 2500 kg/m3) in an air flow at a moderate mass loading of g = 0.5 already gives a particle volume fraction of 0.2
103 but due to inertial effects (gravitational settling or centrifuging) the local solids volume fraction may reach much higher values. Depending on the particle size and pipe configuration interparticle collisions may reduce or enhance wall collision rate as demonstrated by Sommerfeld and Lain [5]. This contribution is aimed at highlighting the influence of wall roughness and inter-particle collisions (i.e. four-way coupling) as well as mass loading on predicted erosion rates using CFD based

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217

APT 2152

No. of Pages 18, Model 5G

26 November 2018 3

S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

number of parcels. Tracking requires the solution of the equations of motion for each computational particle or parcel. The forces which were considered include particle inertia, drag, gravity/buoyancy, slip–shear lift force Fls and slip–rotational lift force Flr [30]. The Basset history term, the added mass and the fluid inertia are negligible for the considered high ratio of particle to gas densities. The change of the angular velocity along the particle trajectory due to the viscous interaction with the fluid (i.e. the torque Ti) requires the solution of additional partial differential equations. Hence, the complete equations of motion for the particles are given by:

234

erosion modelling. It should be emphasized that at present wall roughness is not considered directly in a modified erosion rate (which should be realistically also accounted for) but only in an altered wall collision frequency through the shadow effect [5], whereby naturally erosion rate is altered. The following section (Section 2) gives a brief overview of the Euler/Lagrange method developed, the particle tracking procedure and the different models considered for describing all relevant particle-scale transport processes. Then the considered erosion model is described. As a first step of course the numerical computations including the selected erosion model are validated based on experiments. Then a detailed analysis of the influence of inter-particle collisions, particle size and mass loading on bend erosion is presented. For this purpose a pneumatic conveying system with 5 m horizontal pipe, a vertical bend and a 5 m vertical pipe having a diameter of 150 mm is considered [36]. Finally conclusions are provided as well as an outlook to future work.

235

2. Summary of numerical approach

236

According to the previous considerations, the numerical scheme adopted to calculate the pneumatic conveying system was the fully coupled Euler/Lagrange approach as described in detail by Lain and Sommerfeld [37]. All numerical calculations are conducted by the in-house code FASTEST/Lag3D. The fluid flow was calculated based on the Euler approach using a fixed grid by solving the Reynoldsaveraged conservation equations (RANS) in connection with the standard k-e turbulence model. All conservation equations were extended in order to account for the effects of the dispersed phase, i.e., two-way coupling, through momentum exchange and turbulence modulation (i.e. source terms also appear in the transport equations of the turbulent kinetic energy k and dissipation rate e). The time-dependent three-dimensional conservation equations for the fluid may be written in the general form as:

dxpi Ip ¼ Ti dt

218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233

237 238 239 240 241 242 243 244 245 246 247

248 249

250

252

@ðq/Þ @ðqU i /Þ @ @/ þ ¼ ðCik Þ þ s/ þ s/p @t @xi @xi @xi

ð1Þ

265

where q is the gas density, Ui are the Reynolds-averaged velocity components, and Cik is an effective transport tensor. The first and the second term on the left side of Eq. (1) are the temporal change and the convective term and on the right hand side the diffusion and the source terms appear. The usual source terms within the continuous phase equations are summarised in S/, while S/p represents the additional source term due to phase interaction, i.e. the influence of the particles on the fluid flow. Depending of the variable / = 1, Uj, k, e, the expressions for Cik and S/ can be found in Table 1 of [34]. In the present study, the computations have been done for a stationary flow situation using the steady-state coupling procedure with under-relaxation of source terms described by Sommerfeld [31,34].

266

2.1. Lagrangian particle tracking

267

The simulation of the particle phase by the Lagrangian approach is based on tracking a large number of particles through the beforehand computed flow field. Particles are treated as point-masses and their shape is assumed to be spherical. In order to account for the correct particle mass flow rate, the considered computational particles represent a certain number of real particles with the same properties, which yields a computationally manageable

253 254 255 256 257 258 259 260 261 262 263 264

268 269 270 271 272 273

65

k1 0.12

k2 2.3Hv

k3 0.038

0.19

275 276 277 278 279 280 281 282 283

284

dxpi ¼ upi dt

ð2Þ

286

! !
! dupi 3 q q   mp mp cD ðui  upi Þ u  uP þ mp gi 1  þ Flsi þ Flri ¼ qp 4 Dp dt

287

ð3Þ

289 290

ð4Þ

292

where xpi, upi and ui are the components of the particle location vector and the instantaneous particle and fluid velocities, respectively. Furthermore, mp is the particle mass, Ip the moment of inertia, Dp the particle diameter, gi the gravity vector and q and qp are the fluid and particle material densities. The relevant different forces acting on the particles and the respective resistance coefficients allowing the extension of the equation of motion to higher particle Reynolds numbers have been used according to Sommerfeld et al. [30] as well as Lain and Sommerfeld [34,37] and therefore are not repeated here. The particle motion is computed by integration of the differential equations (Eqs. (2)–(4)). For sufficiently small time steps and assuming that the forces remain constant during this time step, the new particle location, the linear and angular velocities are calculated. The time step for the particle tracking, DtL, was chosen to be 50% of the smallest of all local relevant time scales, such as the particle relaxation time, the integral time scale of turbulence and the mean inter-particle collision time. This choice guarantees the stability of the numerical integration scheme [38,39]. Consequently, particle tracking is done for each parcel independently by adapting the Lagrangian time step automatically according to the local time step limitations.

293

2.2. Modelling of elementary processes

314

In order to account for turbulence effects on particle motion, the instantaneous fluid velocity components along the particle trajectory are determined from the local mean fluid velocity interpolated from the neighbouring grid points and a fluctuating component generated by a single-step isotropic Langevin model described by Sommerfeld et al. [40] and Laín and Sommerfeld [34]. In this model the fluctuation velocity is composed of a correlated part from the previous time step and a random component drawn from a normal distribution function with a standard deviation of the local fluid velocity fluctuation. The degree of correlation depends on the turbulent particle Stokes number StT and is calculated using appropriate time and length scales of turbulence estimated from the k-e turbulence model. Especially in wall bounded flows, as considered here, the modelling of particle-wall collisions needs special consideration. The

315

Table 1 Parameters used in the Oka model for the SiO2 angular particles and aluminium wall. K

274

uref (m/s) 104

Dref (lm) 326

n1

n2 0.14

0.71Hv

2.4Hv0.94

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313

316 317 318 319 320 321 322 323 324 325 326 327 328 329

APT 2152

No. of Pages 18, Model 5G

26 November 2018 4

S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

361

change of the linear and angular particle velocity during a wall collision process is calculated based on the solution of the impulse equations coupled with Coulomb’s law of friction. Separating the process in a compression and rebound period and introducing a restitution coefficient as the ratio of the normal impulse of rebound to that of the compression yields two sets of equations for a sliding and sticking collision. The distinction between these two types of collisions is made on the basis of the friction coefficient. Hence, two model parameters are needed to calculate the wall collision process, which are dependent on impact velocity, particle size, and impact angle and generally have to be extracted from measurements [41]. Corresponding correlations for the restitution coefficient, the friction coefficient and the roughness angle are presented by Lain et al. [42] which are depending on the particle impact angle [2]. In addition, wall roughness was found to have a substantial influence on the particle–wall collision process, depending on particle size and wall roughness structure. Since a detailed consideration of the wall roughness profile in the Lagrangian calculations is not feasible, a stochastic model was developed. For that purpose it was assumed that the instantaneous wall collision angle a is composed of the particle trajectory angle a0 and a stochastic contribution from wall roughness: a = a0 + nDc, where n is a random number with zero mean and variance of 1, and Dc is the standard deviation of the roughness angle distribution depending on particle size and roughness height [41]. From thorough experiments and numerical calculation correlations between Dc and particle diameter with the degree of roughness as a parameter were elaborated and fitted by appropriate correlation functions [32]. The probability distribution function (PDF) of the roughness angle with the standard deviation Dc is described with the following normal distribution function:

364

  1 c2 PðcÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  2pDc2 2pDc2

330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355

356 357 358 359 360

362

389

However, due to the shadow effect this distribution function is effectively skewed to positive values of the roughness angle, especially for smaller wall impact angles [41], since the particles are not able to reach the leeside of the roughness structures. For such cases there is a pronounced transfer of horizontal to transverse particle momentum associated with a strong increase of particle fluctuation velocity and consequently, wall collision frequency in pipe and channel flows. Only if these details are incorporated in the wall collision model can it be expected to correctly predict particle velocities and pressure drop [37]. Inter-particle collisions are modelled by the stochastic approach described in detail by Sommerfeld [43]. This model relies on the generation of a fictitious collision partner at each time step of tracking a real particle. This model is numerically very efficient as it avoids the expensive search of a possible collision partner among neighbouring particles, as required for deterministic collision models. The fictitious particles are representative of the local particle population and hence, their size and velocity are sampled from previously determined local distribution functions. The velocity distributions are assumed to be normally distributed, and the possibility of a correlation between the velocities of colliding particles in turbulent flows is accounted for through a dependence on the particle turbulent Stokes number. Once all instantaneous properties of the real and fictitious particle are known, the collision probability is calculated as:

392

P¼

365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388

390

393 394 395

p 6


!
! ðDpt þ Dpf Þ2  upr  upf 
np Dt L

The subscripts pr and pf refer to the real and fictitious particle, respectively, np is the particle number concentration, and DtL represents the Lagrangian time step of particle tracking. A collision

will occur when a random number in the range [0, 1] becomes smaller than the collision probability. In order to determine the point of impact on the surface of the particles, the velocities (linear and rotational) are transformed in a coordinate system where the fictitious particle is stationary. In this configuration, the relative velocity vector is aligned with the axis of the collision cylinder, and the lateral displacement of the centers of both particles, as well as the appropriate collision angles, are sampled by a random process [43]. By solving again the impulse equations in connection with Coulomb’s law of friction two sets of equations for a sliding and sticking collision are obtained whereby the new velocities of the real particle are calculated. These velocities are then retransformed in the fixed Eulerian frame of reference. Subsequently the velocities of the fictitious particle are not of interest. Full details of the model are given in [43]. It should be also mentioned at this point, that the advanced particle tracking modelling introduced briefly above and implemented in the in-house code FASTEST/Lag3D goes beyond the models used in any available commercial code. During particle trajectory calculation, the particle-wall interaction information such as impact speed, impact angle, and impact location as well as impacting mass is stored. This information is then introduced in the appropriate erosion equations, which relate the particle impact variables with the corresponding erosion, to compute wear in the system which is the last step in the approach. As mentioned before this procedure does at the moment not account for any modification of wall structure or roughness due to erosion.

396

2.3. Two-way coupling procedure

424

Two-way coupling considers the momentum transfer from the dispersed phase to the continuous phase and vice versa through appropriate source terms in the momentum equations and the conservation equations of turbulent kinetic energy and the dissipation rate. The source terms are accumulated for each control volume during the Lagrangian tracking procedure. An underrelaxation approach is used when introducing the source terms in the conservation equations of the fluid flow (for details see Kohnen et al. [44] and Lain and Sommerfeld [1]). When additionally considering inter-particle collisions, whereby the particle-phase properties themselves are modified, the coupling between the two phases is termed four-way coupling. In the cases of two- and four-way coupling the Euler and Lagrange modules have to be run sequentially to get a converged solution, prior to the application of any erosion computation. Initially, the flow field is calculated without particle phase source terms until a converged solution is achieved. Thereafter, a large number of parcels are tracked through the flow field (up to 106 computational particles) and the source terms are averaged for each control volume. In this first Lagrangian calculation interparticle collisions are not considered, since the required cellbased particle phase properties are not yet available. Hence, for each control volume the particle concentration, the local particle size distribution and the size-velocity correlations for the mean velocities and the rms values are ensemble averaged from the traversing particles. These properties are updated during each Lagrangian iteration in order to allow correct calculation of interparticle collisions. Additional particle phase properties and profiles may be collected when the computational particles cross a predefined location. From the second Eulerian calculation, the source terms of the dispersed phase are introduced using an underrelaxation procedure in order to facilitate convergence [1,44]. For the present calculations typically about 25 to 35 coupling iterations with an under-relaxation factor between 0.5 and 0.1 were necessary in order to yield convergence of the Euler-Lagrange cou-

425

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423

426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459

APT 2152

No. of Pages 18, Model 5G

26 November 2018 5

S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

468

pling. The convergence behaviour was clearly demonstrated by Lain and Sommerfeld [1] including two- and four-way coupled simulation periods. All the numerical computations were done on a quad-core workstation, where the fluid flow was calculated on one processor and the Lagrangian tracking on four processors. With these resources, a four-way coupled converged result for the elbow configuration was obtained after about two weeks with the in-house code FASTEST/Lag3D.

469

3. Erosion estimation

470

506

The mathematical description or correlations of erosion rates has been subject of research at least since the work of Finnie [45]. Meng [46] and Meng and Ludema [47] conducted a literature survey over more than five thousand papers and found hundreds of equations for prediction of friction and wear, the great majority for specific conditions and not applicable beyond the range of parameters under which they were developed. From those studies, it was concluded that the empirical models were valid only within the range of the experiments but they were more accurate in that range than theoretical equations. Further analyses of theoretical erosion models accuracy gave best predictions ranging from 0.4 to 2.7 times the value of measurements [10], or within an order of magnitude of measured results. Besides, the literature strongly supports the notion that to predict good quality erosion rate data it is essential that rates be determined empirically by experiment using the same surface and impacting materials as expected in the engineering application of interest [48]. After a literature review looking for the most recently cited erosion models, there are two of them with far more cites than the rest: the E/CRC model of the Erosion Corrosion Research Center of the Tulsa University (in different versions) and the model developed by Oka et al. [3] and Oka and Yoshida [18]. Both models were extensively compared by Zhang et al. [10] in direct impact experiments in both liquid-solid and air-solid flows. Good agreement with experiments was found for both models. It is remarkable that one of the authors, Y. Zhang, former member of the E/CRC group, has adopted the Oka model in later publications (e.g., [49]) instead of the E/CRC model. Moreover, one of the purposes of the Oka et al. [3] paper was ‘‘to propose predictive equations for erosion damage caused by solid particle impact that can be applied to many types of metallic materials under various conditions involving impact angle, velocity, size and the properties of the particles”. This fact makes such a model particularly attractive and, consequently, it has been adopted in this work. Therefore, the Oka model is briefly summarized in the following. The predictive equation for volumetric erosion damage (mm3/ kg erodent agent) proposed by Oka et al. [3] can be expressed as:

509

EV ¼ K ða H v Þ

460 461 462 463 464 465 466 467

471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505

507

510 511 512 513 514 515 516 517 518 519 520

521 523

 k1 b

up uref

k2 

gðaÞ ¼ ðsinaÞ ½1 þ Hv ð1  sinaÞ n1

Dp Dref

k3

gða; Hv Þ

ð5Þ

n2

ð6Þ

n1 and n2 are exponents determined by the eroded material hardness and other impact conditions such particle properties and shape. These exponents show the effects of repeated plastic deformation and cutting action; they are expressed by:

n1 ¼ s1 ðHv Þq1 ;

This expression depends on the properties of wall material through the Vickers hardness Hv and particle properties (shape and hardness) collected in the parameter K. In Eq. (5), up and Dp are the impact velocity (m/s) and particle diameter (lm), respectively, and uref and Dref are the reference impact velocity and particle diameter used in the experiments [3]. a and b are parameters that characterize the load relaxation ratio of the wall material, while the exponents k1 and k3 are determined by the properties of the particle, although k3 it is found to be roughly constant in the experiments. The velocity exponent k2  2 is correlated with the eroded wall material hardness and particle properties by:

k2 ¼ rðHv Þp

The function g(a) represents the impact angle dependence of erosion rate, expressed by two trigonometric functions and by the initial eroded material Vickers hardness number (Hv) in units of GPa, as:

n2 ¼ s2 ðHv Þq2

In Eq. (6) the term (sin a) accounts for erosion by deformation (close to normal impact) and the term ½::::n2 stands for the peeling action at low impact angle. From a theoretical point of view, an attractive feature of the Oka model is that for a particular combination of eroded material and eroding agent, the coefficients can be obtained from more fundamental properties which are measurable such as the Vickers hardness. Finally, it is necessary to remark that the vast majority of CFD based erosion modelling has been compared with experiments performed under conditions of very low particle mass loading. These operating conditions ease the measurements, on one side, and on the other the computations are much faster because the coupling between the Euler and Lagrange modules is avoided. Moreover, the validation of CFD based erosion models in complex geometries such as bends, tees, valves . . . has as target variable an erosion measure (erosion rate, penetration ratio, . . .) along the experimental specimen, but almost never the flow field is validated. Therefore, in order to properly validate CFD erosion models, there is a need of conducting erosion experiments where both, erosion rate and fluid and particle variables, are measured.

524 525 526 527

528 530 531 532 533 534

535 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558

4. Results

559

4.1. Validation of erosion computations

560

Validation of the erosion model in this work has been performed by comparing the numerical results with the vertical to horizontal bend flow of the Mazumder et al. [4] experiments. Aluminum (6061-T6) elbow specimens (density qw = 2700 kg/m3, Hv = 1.049 GPa) were used in these experimental studies. By applying CFD-based models, erosion predictions were performed in the same geometry. The continuum phase consists on air with a bulk velocity of 34.1 m/s, corresponding to a Reynolds number of about 57,750. Angular sand particles (density 2600 kg/m3), with a mean diameter of 182 lm and a mass loading of 0.013 (kg particles/kg air), were injected vertically at about 1.22 m below the test piece. The test piece was located on the outer wall of a 90° elbow with a diameter of D = 0.0254 m and a curvature radius of 0.0381 m, as shown in Fig. 1. In order to represent more realistic behaviour at the specimen location regarding the particle properties, the whole domain was included on the calculation. After the elbow a horizontal pipe which is 10 diameters long is considered. This configuration attempts to reproduce better the experimental conditions. Since the geometry is simple, only hexahedra were used in the structured mesh construction, which generate less numerical diffusion than elements such as tetrahedral and pyramids. Gradual refinement is necessary in the near-wall region, where high velocity gradients and boundary layers exist. Hexahedra also allow the generation of higher quality meshes, with fewer distorted elements than one would obtain with tetrahedral, for example. The mesh resolution used in all simulations is approximately 1,000,000 elements, which yielded mesh-independent results. At

561

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587

APT 2152

No. of Pages 18, Model 5G

26 November 2018 6

S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

Fig. 2. Dependence of the function g(a) on the particle–wall impact angle for different wall hardness and particle types: the case of Mazumder et al. [4] and the case of Huber and Sommerfeld [36].

Fig. 1. Sketch of the experimental configuration of Mazumder et al. [4]; (vertical pipe length 48 D = 1.22 m, horizontal outlet pipe length 10 D = 0.254 m).

588 589 590 591 592 593 594 595 596 597 598 599 600 601

602 604

the inlet, 48 diameters upstream the bend, a fully developed air flow based on a 1/7-potential law is imposed. Particles are introduced with the mean fluid velocity of 34.1 m/s and a fluctuating velocity for all components of around 4.5% of the bulk flow velocity was specified. The experiments were performed with a very low mass loading of 0.013, so inter-particle collisions as well as flow modification by the solids are disregarded in the computations. Initially, the duct walls were considered perfectly smooth. In the simulations the restitution model of Grant and Tabakoff [50] was chosen because such model was originally obtained from experiments with sand particles hitting against an aluminium surface, the same materials than used in the Mazumder et al. [4] experiments. The normal restitution coefficient en and its standard deviation ren are given by (the impact angle a is given in radians):

en ¼ 0:993  1:76a þ 1:56a2  0:49a3

ren ¼ 2:15a  5:02a2 þ 4:05a3  1:085a4

607

The employed mean friction coefficient, depending on the particle-wall angle of impact, is the same as in the Laín and Sommerfeld [1,37] publications. It is written as

608 610

l ¼ maxð0:15; 0:5  0:0175aÞ

605 606

611 612 613 614 615 616 617 618 619 620

ð7Þ

where a is again the particle impact angle, but this time in degrees. In the present steady state computations, 106 particles are tracked through the computational domain using the forces and time step restrictions described earlier. The parameters of the Oka et al. [3] model for estimating the erosion due the sand (SiO2) angular particles on the aluminium specimen, which were the materials used in the experiments [4], are listed in Table 1. As shown in Fig. 2, the function g(a) of Eq. (6) for the Mazumder et al. [4] case, rapidly increases reaching a maximum around 25°

wall impact angle and finally approaches a value of one for 90°. This behaviour is due to the rather soft wall material and indicates that cutting is the dominating erosion mechanism in this configuration. Fig. 3a presents the predicted erosion pattern in terms of erosion depth or thickness loss on the outer wall of the bend of Mazumder et al. [4]. The thickness loss is obtained as follows: the erosion damage [mm3 eroded material/kg erodent agent], Eq. (5), is first converted into penetration ratio [m eroded material/ kg erodent agent] through the process described by Eqs. (8)–(10) below and then multiplying the result by the mass of injected particles the thickness loss [m eroded material] is computed. In the case of [4] the injected particle mass was 1 kg. The flow direction is upwards, so the gravity is directed in the opposite direction of the flow in this figure. The obtained erosion pattern is very similar to that obtained by other authors using CFD-based erosion models in bends [9,15,27,54,55]. It consists of a narrow elongated spot of high thickness loss, due to primary, high velocity, particle-wall impacts, surrounded by a wider area of moderate erosion. Downstream of the spot of the highest thickness loss, the particle erosion pattern shows a V shaped structure. This last area is mainly produced by secondary particle-wall collisions in the elbow [9]. However, following Soldornal et al. [51] this erosion pattern never appears experimentally. As it is illustrated below, such a texture arises because wall roughness is neglected when computing particle-wall interactions. A quantitative comparison of the numerical results with the measured thickness loss (including error bars) along the central line of the bend outer wall is presented in Fig. 3b. Due to the stochastic nature of particle tracking the calculated erosion depth is an average over a width of 5 mm. Here, the origin (h = 0°) is set at the elbow inlet, and h = 90° corresponds to the elbow flow outlet. It is obvious that the simulation results without wall roughness (solid line) are lying within the lower part of the experimental uncertainty (vertical bars indicating the standard deviation of the measurement), and hence, a reasonable agreement with the experiments is obtained. However, in practice wall roughness is always present, even under erosion conditions. Therefore, also a calculation with a low wall roughness was conducted, showing a remarkable reduction of erosive thickness loss in the middle outer wall region of the bend, i.e., between about 45° to 65°. This issue will be further discussed below, as well as a modification of the friction coefficient. It is known that wall roughness plays a very important role in the particle motion along ducts and strongly affects the wall colli-

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665

APT 2152

No. of Pages 18, Model 5G

26 November 2018 S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

Fig. 3. (a) Predicted erosion pattern on the bend outer wall for the Mazumder et al. [4] configuration (flow direction is upwards); (b) Comparison of the numerical thickness loss with the experimental data (calculated results are an average over a width of 5 mm along the bend outer wall, the experimental error bars indicate the standard deviation of the erosion loss thickness).

666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688

sion frequency [1,5]. It should be noted that no information about wall roughness was available from the experiment. Therefore, additional cases with increasing roughness were simulated (Dc = 1°, 2.5°, 5°, 10° in the notation of Laín and Sommerfeld [1]) to illustrate its effect on erosion computations. Fig. 4 illustrates the variation of the erosion pattern with increasing roughness. As roughness augments, the maximum erosion decreases, the maximum thickness loss spot blurs and the downstream V shaped pattern tend to disappear. For high enough roughness the predicted erosion pattern shows an ellipsoidal-like shape, in agreement with the experimental and numerical results of Soldornal et al. [51]. Such behaviour can be explained because wall roughness redisperses inertial particles remarkably and the dust rope becomes much wider [5], which reduces the near wall particle concentration and eventually results in a reduction of the wall collision frequency. Also, the kinematic properties of the wall colliding particles, velocity and impact angle, are modified by wall roughness (see Fig. 5). To quantitatively illustrate the effect of wall roughness, Fig. 5 shows the profiles of different variables along the central line of the bend outer wall (width 5 mm), where the bend angle h ranges from 0° to 90°. It is obvious from Fig. 5a that the maximum of thickness loss is remarkably reduced by wall roughness and also

7

that the location of the erosion peak is slightly displaced toward lower bend angles. The thickness loss obtained for moderate wall roughness (Dc = 2.5°), represented by the dashed line, has been included in Fig. 3b to be compared with the case without roughness. In this case, the peak of erosion has been reduced around a 35% regarding the smooth wall case. The effect of wall roughness, the consequent particle re-dispersion and the reduction of particlewall collision frequency reflects on a remarkable reduction of the average particle-wall impact velocity (i.e. magnitude of the absolute component, see Fig. 5b). As particles are rebounding at larger angles due to roughness this yields also an increase of the average impact angle (the impact angle is defined as the angle between particle trajectory and the tangent plane to the wall at the collision point) along the bend (Fig. 5c) and a noteworthy decrease of the particle impacting mass (Fig. 5d) in the range of bend angles between 30° and 75°. All these trends imply lower values of the volumetric erosion damage in the Oka model. It is interesting to notice that similar tendencies for impact particle velocity, impact particle angle and thickness loss with increasing roughness were obtained by Duarte et al. [25] who compared their predictions with the experiments of Soldornal et al. [51]. Finally, the influence of the friction coefficient l is investigated in Fig. 6, where simulation results neglecting wall roughness are presented. Four friction coefficients are considered: that one presented in Eq. (7) variable with particle impact angle, and constant values of 0.1, 0.175 and 0.5. From Fig. 6a it becomes evident that thickness loss decreases as the friction coefficient increases. Moreover, in Fig. 3b, the dot-dashed line corresponding to the constant value of l = 0.175 is included, which shows that the maximum of erosion rate increases regarding the basic case (solid line), which has an average value around of 0.25. The results for thickness loss are reduced by a 40% when friction coefficient is doubled to a value of 0.5, suggesting a non-negligible sensitivity to this parameter. Such trend can be explained because higher friction implies a decrease in particle velocity (i.e. mainly the wall parallel component) regarding the cases with lower friction (which can be confirmed in Fig. 6b) and, as the erosion ratio varies with particle impact velocity to a power larger than two, higher friction implies lower erosion and vice versa. As a consequence, average particle impact angle raises and particle impacting mass decreases along the bend outer wall when friction coefficient increases (see Fig. 6c and d). It is interesting to comment that such trend is different from that found by Pereira et al. [9], in the same elbow configuration, where the erosion rate was not appreciably dependent of the constant friction coefficient employed in the simulation. However, in a later paper the same authors, Duarte et al. [25], find the same trends of behaviour of erosion rate regarding the friction coefficient that those explained in the present work. Therefore, in any case, the applied wall collision model and the involved parameters have to be selected correctly in order to allow a proper prediction of erosion rates.

689

4.2. Application to the elbow flow of Huber and Sommerfeld [36]

740

The paper of Mazumder et al. [4] does not provide information about the gas or particle phase variables, i.e. particle concentration and velocities of both phases, so it is not possible to validate the predicted two-phase flow field. For such reason, a complementary analysis of bend erosion has been conducted in the configuration of a 5 m horizontal pipe, a bend, and a 5 m vertical pipe with a pipe diameter D = 150 mm. The flow in this configuration was experimentally studied by Huber and Sommerfeld [36], case 3, and numerically computed by Laín and Sommerfeld [34] and Sommerfeld and Lain [5,24] showing a good agreement in all available variables. Hence, all the models describing the particle transport in pipe systems, i.e. turbulent transport, collisions with rough walls

741

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739

742 743 744 745 746 747 748 749 750 751 752

APT 2152

No. of Pages 18, Model 5G

26 November 2018 8

S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

Fig. 4. Variation of the erosion pattern with increasing wall roughness: (a) Dc = 1°, (b) Dc = 2.5°, (c) Dc = 5°, (d) Dc = 10°; Mazumder et al. [4] case; flow enters from the lower part.

753 754 755 756 757 758 759 760 761 762 763 764

and inter-particle collisions were validated thoroughly. However, this pipe system has a bend with larger diameter, i.e. larger bend radius, than the Mazumder et al. [4] configuration, namely, Rbend = 2.54 D, and lower conveying velocity so that bend erosion is expected to be much smaller. Other characteristics of the Huber and Sommerfeld [36] elbow flow are the following: air bulk velocity 27 m/s; stainless steel pipe material (which is characterized by a pretty high roughness, realized in the numerical computations by a wall roughness parameter of Dc = 10°); particle phase consists of spherical glass beads (qp = 2500 kg/m3) with a number mean diameter of about 40 lm and mass loading ratio g = 0.3. The particle size distribution is shown in Fig. 7, which numerically was dis-

cretized in 7 diameter classes in the range (20, 80) microns with a width class of ten microns. The computational domain of the horizontal to vertical elbow was discretized by a multi-block structured grid with 25 blocks and a total number of 568,000 hexahedral cells which were found sufficient to reach grid independent results. Particles were injected uniformly at the inlet with stream-wise velocities sampled from a Gaussian distribution with a mean value corresponding to the bulk fluid velocity (i.e., 27 m/s) and a rms (root mean square) value of 5% of this velocity. The restitution and friction coefficient models employed in present computations for describing particle-wall interactions are those given in Laín and Sommerfeld [1].

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

765 766 767 768 769 770 771 772 773 774 775 776

APT 2152

No. of Pages 18, Model 5G

26 November 2018 S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

9

Fig. 5. Effect of wall roughness on different variables along the central line of the bend outer wall (width 5 mm); (a) thickness loss; (b) magnitude of the absolute, average particle impact velocity; (c) average particle impact angle regarding the tangential plane to the wall at the particle impact point; (d) impacting particle mass; Friction coefficient given by Eq. (7); Mazumder et al. [4] case.

777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799

Apart from the aforementioned differences between this test case and that of the Mazumder et al. [36] configuration, another one is the gravity direction which is perpendicular to the horizontal pipe in this case. Therefore, gravitational settling produces a non-symmetric particle concentration distribution at the bend inlet. This will of course yield a modified erosion pattern at the bend outer wall. Moreover, the Huber and Sommerfeld [36] case, considers spherical glass beads instead of angular sand particles with smaller size and conveying velocity, yielding of course also lower erosion rates. As a consequence of the long horizontal pipe, when particles reach the bend entrance they show a fairly non-uniform concentration profile, which is illustrated in Fig. 8 where a cut through the elbow mid-plane is shown. This fact favours the occurrence of inter-particle collisions. As already discussed in Lain and Sommerfeld [34] inter-particle collisions induce a compression of the powder rope formed in the bend altering the particle wall collision frequency. This is illustrated in Fig. 9 where the cross-sectional maxima of the particle concentration normalised by the mean value are plotted along the distance, normalised by the pipe diameter. Due to particle inertia, the maximum concentration is reached close to the elbow exit and has a value which is more than 30 times higher than the mean concentration for the case of two-way cou-

pling (Fig. 9). When considering inter-particle collisions the resulting compression effect yields a drastic increase of maximum concentration to about 100 times the mean value, also just before the bend exit. In the connecting vertical pipe the maximum concentration again decreases due to particle dispersion. As it was mentioned in the introduction, the wall collision rate strongly depends on pipe configuration and particle size, which is shown in Fig. 10. For each pipe element the wall collision rate is shown for two- and four-way coupled calculations for particle sizes ranging from 20 lm to 135 lm (note that here mono-sized spherical glass beads are considered). Expectedly, the highest wall collision rate is found in the bend. For all pipe elements the effect of inter-particle collisions is completely different for particles below and above 30 lm under these conditions. Above 30 lm the wall collision frequency is remarkably increased due to collisions between the particles within the bend and the vertical pipe. This is caused by the fact that particles entering the bend are colliding with those being rebound from the bend outer wall. Thereby the concentrated particle region (called dust rope) is pushed towards the bend outer wall (compression effect) and a much denser rope is formed with inter-particle collisions as shown in Figs. 8 and 9 [5]. This trapping effect implies that particles are strongly bouncing with other particles in the rope when this is moving along

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822

APT 2152

No. of Pages 18, Model 5G

26 November 2018 10

S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

Fig. 6. Effect of friction coefficient on different variables along the central line of the bend outer wall; (a) thickness loss; (b) magnitude of the absolute, average particle impact velocity; (c) average particle impact angle regarding the tangent plane to the wall at the particle impact point; (d) impacting particle mass; Computations without wall roughness; Mazumder et al. [4] case.

Fig. 7. Size distribution of spherical glass beads (relative number frequency) used in the experiments of Huber and Sommerfeld [36].

823 824 825 826 827 828 829 830

the bend and thereby very frequently will collide with the walls. This effect is reduced when increasing particle size and hence inertia due to the falling particle number density at the same volume fraction. Of course, such modification of particle-wall collision frequency will alter the erosion rate. Particles smaller then about 30 lm show the opposite trend, namely reduction of wall collision rate through inter-particle collisions. However, also for these small particles the rope forming in

the bend is being compressed through inter-particle collisions. Hence, the reason for the reduced wall collision rate must be sought in a different physical phenomenon. The turbulent Stokes number for particles of about 37 lm is around unity with respect to the integral time scale in the core of a pipe flow [5]. This implies that the velocity of particles below this size becomes more and more correlated which will reduce the inter-particle collision rate and as a consequence also the compression effect. Also after a collision with other particles in the rope they will not bounce back onto the bend outer wall, as they are more easily carried along the bend with the main flow. Consequently, a reduction of wall collision rate in the bend and the vertical pipe through inter-particle collisions is found for such small particles. For the estimation of erosion in the configuration of Huber and Sommerfeld [36] using the Oka model, the Vickers hardness of the wall made of stainless steel has been taken as Hv = 1.96 GPa, [49], with a density qw = 7929 kg/m3. Fortunately, Oka et al. [3] and Oka and Yoshida [18] provided the combination of materials of stainless steel and glass beads; therefore the proper parameters for the erosion computations are known and they are listed in Table 2. The dependence of the parameter g(a) of Eq. (6) on the impact angle for stainless steel is also shown in Fig. 2. Due to the larger hardness of stainless steel the material may be classified as brittle and as a consequence cutting erosion at small impact angles is zero. As a result the parameter g(a) increases continuously from zero to unity for impact angles from 0° to 90°.

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856

APT 2152

No. of Pages 18, Model 5G

26 November 2018 S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

11

Fig. 9. Normalised maximum particle concentration (i.e. normalised by the mean value) along a part of the horizontal pipe, the bend and a part of the vertical pipe (i.e. distance normalised by pipe diameter) showing the influence of inter-particle collisions (i.e. two- versus four-way coupling); the maximum particle concentration is obtained for each cross-section along the shown pipe segment, in the bend this maximum is close to the bend outer wall (case with particle size distribution from 15 to 85 lm, configuration: 5 m horizontal pipe, bend, 5 m vertical pipe, bend radius 2.54
D, D = 150 mm, Uav = 27 m/s, g = 0.3, Dc = 10°).

Fig. 8. Particle concentration in kg/m3 in the elbow mid-plane; (a) two-way coupling; (b) four-way coupling; (case with particle size distribution from 15 to 85 lm, D = 150 mm, bend radius 2.54 pipe diameters, Uav = 27 m/s, g = 0.3). Arrow indicates flow direction.

857 858 859 860 861 862

The erosion results are presented in terms of the penetration ratio (PR) which expresses depth of eroded wall material over mass of erodent agent, i.e., m/kg. For that, the erosion damage in mm3/kg provided by the Oka model must be converted in penetration ratio in the following way: first the erosion damage is converted in erosion ratio ER (kg eroded material/kg erodent agent) by means of:

863 865 866 867 868

ER ¼ 109 qw EV ðaÞ

ð8Þ

where the constant 109 has the dimension [m3/mm3]. Then, the penetration ratio PRS [m/kg] at each surface element of area AS is obtained as (the index S marks the surface area):

869 871 872 873 874 875

PRS ¼

ERS qw AS

ð9Þ

where ERS is the erosion ratio in the surface element. It is computed as the sum of erosion ratio caused by NS particle-wall collisions in the surface area AS [15] divided by the total injected mass of erodent agent mT:

Fig. 10. Dependence of wall collision frequency on particle size (obtained from simulations with mono-sized particles) showing the influence of inter-particle collisions (i.e. two- versus four-way coupling); closed lines and closed symbols: four-way coupling; dashed lines and open symbols: two-way coupling (5 m horizontal pipe, bend, 5 m vertical pipe, bend radius 2.54
D, D = 150 mm, Uav = 27 m/s, g = 0.3, Dc = 10°).

PNs  ERS ¼

i¼1

mpi ERi mT

876

 ð10Þ

In Eq. (10), mpi represents the mass of particle i (or better parcel i) colliding with the wall. The penetration ratio obtained with the Oka et al. [3] model is illustrated in Fig. 11 using colour contours. Note that for these simulations a wall roughness parameter of Dc = 10° has been considered due to pretty high roughness of stainless steel. Without particle–particle collisions, the area of maximum erosion has an elongated tear-like shape which peak is located at a bend angle of around 35° from the inlet. It is interesting to note that the Vshaped region appearing in Fig. 3a for the narrow bend of Mazumder et al. [4] is not present. Beyond h = 50° the erosion is rather weak. When inter-particle collisions are considered, i.e. four-way coupling, the peak value of penetration ratio decreases more than 30%, the location of maximum erosion is displaced toward smaller

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

878 879 880 881 882 883 884 885 886 887 888 889 890 891 892

APT 2152

No. of Pages 18, Model 5G

26 November 2018 12

S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

Table 2 Parameters used in the Oka model for the glass beads and stainless steel wall. K 27

k1 0.16

k2 2.1

k3 0.19

Fig. 11. Penetration ratio in m/kg in the bend; (a) two-way coupling; (b) four-way coupling; (size distribution 15–85 lm, D = 150 mm, bend radius 2.54 pipe diameters, Uav = 27 m/s, g = 0.3). Flow enters from the lower part. Note the different colour scaling in the parts (a) and (b).

893 894 895 896 897

bend angles (i.e., h  30°), and at higher bend angles only a strip of moderate erosion is observed. The extent of the maximum erosion region in the stream-wise direction is slightly reduced and broadened sideways. This observation for the erosion pattern is strongly linked with the compression of the dust rope by inter-particle col-

uref (m/s) 100

Dref (lm) 200

n1 2.8Hv

n2 0.41

2.6Hv1.46

lisions and a slight lateral widening (see Fig. 8). The reduction of penetration rate in the four-way coupling case is not explained readily, since the particle–wall collision rate is slightly increased due to inter-particle collisions, which will in the first instance enhance erosion rate. Therefore, a decrease in erosion can only result from reduced particle impact velocities or modified impact angles [24,52], which are both included in the erosion model, and hence, their influence will be further discussed in the following. The profiles of penetration ratio and particle properties along the bend (Fig. 12) are obtained by averaging over all impinging particles for a strip of 5 mm along the bend outer wall. Results are compared for mono-sized spherical glass beads of 40 lm and the size distribution (SD) of Fig. 7 with the same number mean diameter without and with inter-particle collisions. First of all it is obvious that inter-particle collisions reduce erosion although the wall collision frequency is enhanced (compare two- and fourway, Fig. 10). Hence, other properties such as particle impact velocity and impact angle must be also modified due to inter-particle collisions. The modifications of these properties were first analysed in detail by Sommerfeld and Lain [24] and will be further elaborated here and the different terms in the erosion model will be analysed. Higher particle wall collision frequency however also implies more momentum loss for the particle phase and consequently lower averaged particle velocity. This velocity reduction is very pronounced for the mono-sized particle (Fig. 12b) but only marginal for the case with size distribution; on the contrary with inter-particle collisions the modulus of impact velocity increases up to bend angles of about 30°. Since the particles are captured within the dust rope due to inter-particle collisions, this is connected with a reduction of the averaged wall impact angle as obvious from Fig. 12c) for the 40 lm-particles. This impact angle reduction begins at a bend angle of about 20° and continues until the bend outlet. For stainless steel, the function g(a) in the erosion model (see Eq. (6)) decreases continuously when the impact angle is reduced (brittle erosion), see Fig. 2. Hence, more shallow impact angles as well as reduced impact velocity yield this remarkable reduction of penetration ratio through inter-particle collisions (Fig. 12a), although wall impact frequency increases. The same fact can also be appreciated in Fig. 13 for larger particles of 135 lm, where inter-particle collisions reduce the PR (Fig. 13a) due to the reduction of particles impact velocity and angle (Fig. 13b and 13c). However, Fig. 13a) clearly shows that the PR generated by the larger particles is much higher than for the 40 lm particles owing to larger particle impact velocity and angle shown in Fig. 13(b) and (c). It should be noted that in this case PR increases with an exponent k2 larger than the square of the wall impact velocity (Table 2). For the case with particle size distribution the interpretation of the results is not so straightforward. A remarkable reduction of penetration ratio due to inter-particle collisions is observed between 20° and 60° bend angle. The impact velocity is however only slightly reduced for bend angles larger than 30° (Fig. 12b). The impact angle, and so the function g(a), are however increased throughout the bend (Fig. 12c). The answer to the nevertheless reduced erosion as a consequence of inter-particle collisions is the existence of a particle size distribution including smaller and larger particles around the mean value. All of these particles are

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956

APT 2152

No. of Pages 18, Model 5G

26 November 2018 S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

Fig. 12. Effect of considering different particle size distributions on bend erosion (mono-sized versus size distribution of Fig. 6) and comparison of two- and four-way coupled simulations; (a) penetration ratio in m/kg in the bend for glass beads particles; (b) averaged magnitude of particle wall impact velocity; (c) average particle wall impact angle regarding the tangent plane to the wall at the particle impact point along the bend outer wall for a 5 mm strip (D = 150 mm, bend radius 2.54
D, Uav = 27 m/s, Dc = 10°, g = 0.3).

957 958 959 960

of course included in the statistics presented in Fig. 12. As demonstrated by Fig. 13, the impact angle and the impingement velocity for larger particles are remarkably reduced by inter-particle collisions [24,52]. This will of course reduce the overall penetration

13

Fig. 13. Effect of particle diameter on bend erosion (mono-sized particles) and comparison of two- and four-way coupled simulations; (a) penetration ratio in m/ kg in the bend for glass beads particles; (b) averaged modulus of particle wall impact velocity; (c) average particle wall impact angle regarding the tangent plane to the wall at the particle impact point along the bend outer wall for a 5 mm strip (D = 150 mm, bend radius 2.54
D, Uav = 27 m/s, Dc = 10°, g = 0.3).

ratio as shown in Fig. 12(a). The contribution of particles smaller than the mean diameter (i.e. 40 lm) to erosion is expectedly lower. In conclusion, although the particle-wall collision frequency is increased due to inter-particle collisions for mono-sized particles larger than about 30 lm (see Fig. 10) the erosion damage is how-

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

961 962 963 964 965

APT 2152

No. of Pages 18, Model 5G

26 November 2018 14 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030

S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

ever reduced, mainly due to the distinct reduction of averaged particle-wall impact velocity and average wall impact angle. A second important finding is that the case of considering a size distribution having the same number mean diameter, yields much higher erosion depth (Fig. 12) in the bend angle range from 10° to 60°. Hence, although fewer particles larger than 40 lm are in the spectrum the overall erosion is drastically increased. For the same reason, erosion also begins at smaller bend angles, as these larger particles have higher inertia and do not easily follow the flow deflection. This implies that mono-sized calculations cannot be used to properly calculate erosion rates compared to experiments which always have a size distribution. Finally, the effects of mass loading on erosion along the elbow are discussed. As an illustration, Fig. 14 shows the evolution of the erosion pattern as particle mass flux is increased, corresponding to g = 0.3 (a), g = 0.6 (b) and g = 1.0 (c), for the four-way coupled case. Due to the increasing particle number density for higher mass loading, an increase in PR and erosion could be expected. However, as it can be readily seen the maximum value of penetration ratio (PR) decreases with increasing g and displaces towards lower values of the bend angle. Moreover, the zone of maximum erosion also changes its shape from a concentrated spot to a sort of U-like shape similar to a crescent moon, surrounded by a tearlike shaped region. Toward the end of the bend, for h > 45°, a narrow strip of higher PR is observed; here the values of PR increase slightly with increasing mass loading ratio. Fig. 15a presents the results of PR for the three particle mass loading ratios g = 0.3, 0.6, 1.0 in a 5 mm wide strip along the bend outer wall which allows a quantitative comparison. It becomes evident that penetration rate decreases with increasing mass loadings for both, two- and four-way coupling. The reduction of the erosion damage due to inter-particle collisions has been explained earlier. Now, the question is why an increased mass loading reduces PR for major parts of the bend outer wall considering both two- and fourway coupling. When inter-particle collisions are neglected a slight decrease of PR with increasing mass loading is observed only between 20° < h < 45°. This observation is explained because with increasing mass loading, the particle rope slows down the fluid in the proximity of the outer wall which, in turn, slows the velocity of particle rope, enhancing the two-way momentum coupling. As a result, the peak of penetration ratio slightly decreases when mass loading increases and also marginally moves to larger bend angles. In the four-way coupling case, however, PR is remarkably reduced with increasing loading between 20° < h < 45° and the maximum is remarkably shifted to smaller bend angles (i.e. from 30° to about 20°). This effect is associated with the stronger gravitational settling at higher mass loading in the horizontal pipe whereby just after entering the bend a concentrated and lateral wide rope develops [5]. In the outflow part of the bend (i.e. beyond about 50° bend angle), however, erosion increases with mass loading and, in the case of g = 1.0, its value is even slightly higher than the first peak present at h  20°. Specifically at this high mass loading the erosion profile is characterised by two maxima (i.e. at 20° and 60°) and a minimum in between (i.e. at 35°). The increase of erosion in this final part of the elbow with growing mass loading is mainly due to the increase of particle-wall collision frequency, as in such area average particle impact velocities and angles decrease with increasing g (see Fig. 15b and c). As demonstrated by Sommerfeld and Lain [5], the dust rope at higher mass loading becomes more concentrated occupying a smaller region of the bend cross section. This means that at higher loading, particles moving into the bend will collide more probably with particles in the concentrated dust rope and not with the bend outer wall. This phenomenon is also connected with the minimum in the penetration ratio at high mass loading and may be called ‘‘shielding” effect, which however is

only relevant for the initial part of the bend [53]. The extent of this region is correlated with particle inertia. The main reason for the reduction of erosion with increasing mass loading (at least for the considered values) is related to the trapping of the particles in the rather dense dust rope moving along the bend outer wall, which increases first the wall collision frequency and thereby also reduces particle velocity due to wall collision momentum loss [24,52]. Moreover, with increasing mass loading momentum coupling in the rope becomes stronger yielding a further reduction of air velocity in this region of a horizontal to vertical bend. As particles are held in this low-velocity region of the gas flow, this naturally involves a reduced average particle– wall collision velocity as summarized in Fig. 15b. A remarkable reduction of the impact velocity with growing mass loading is found from bend angles larger than 20° when comparing the four-way coupled calculations. These arguments are also supported by the fact that the two-way coupled calculations show almost no difference for the considered mass loadings. As a result of the compression of the dust rope and the resulting particle trapping, also the average impact angle is reduced when considering four-way coupling (Fig. 15c). As a consequence, the average impact kinetic energy is also reduced for bend angles higher than 20° as mass loading increases (Fig. 15d), which is the result of particle momentum loss and gas velocity decreasing due to coupling. In the dilute conveying regime reduced erosion may be observed with increasing mass loading for a horizontal to vertical bend due to inter-particle collisions damping particle motion within the compressed rope. As a final comment, it is necessary to emphasise that decreasing values of specific erosion measures (i.e. penetration ratio [mm/kg]) with growing mass loading not necessarily imply lower erosion rates (e.g., penetration rate [mm/hr]; thickness loss as a function of time), which is more useful for continuously running processes. For instance, in the cases presented in Fig. 15a, the penetration rate, obtained as the penetration ratio times the particle mass flow rate, becomes larger for growing mass loadings, i.e., the erosion velocity increases with increasing particle loading as shown in Fig. 16. All the curves show similar trends as for the penetration ratio (Fig. 15a) only that the penetration rate increases with growing mass loading. These two trends are summarised in Fig. 17 where the total erosion ratio [mm3/kg] and the total erosion rate [mm3/s], both in the entire bend, are plotted versus the particle mass loading. As expected from Fig. 16, the erosion rate (temporal loss of bend volume) increases with mass loading for both two-way and four-way coupling; however, it is remarkably lower when considering interparticle collisions due to the shielding effect. As discussed above, no matter which erosion parameter is used, inter-particle collisions cause a reduction of erosion whose difference increases with mass loading due to the growing importance of inter-particle collisions. The total erosion ratio in [mm3/kg] shows different trends (Fig. 17). For two-way coupling only a small variation of erosion volume with mass loading is observed, yielding 7% less erosion at g = 1.0 compared to g = 0.3. In contrast, if accounting for inter-particle collisions, a clearly visible decrease of the erosion ratio with growing mass loading is observed, which is 35% from g = 0.3 to g = 1.0. Finally, the volume erosion ratio is analysed with respect to particle size, considering only mono-sized particles at a mass loading ratio of g = 0.3 (Fig. 18). The erosion increases almost linearly with particle size when neglecting inter-particle collisions. This is because of two competing effects, first the particle number density 3

decreases with growing particle size which is proportional to dP . On the other hand the impact energy increases with the cube of the particle diameter yielding higher erosion. However, as Fig. 13 b and c clearly show, both, the particle impact angle and velocity

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095

APT 2152

No. of Pages 18, Model 5G

26 November 2018 S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

15

Fig. 14. Shape of erosion patterns for different mass loadings: (a) g = 0.3; (b) g = 0.6; (c) g = 1.0. Four-way coupled simulations (size distribution 15–85 lm, D = 150 mm, bend radius 2.54
D, Uav = 27 m/s, Dc = 10°).

1096 1097 1098

increase with particle diameter. As a consequence a linear increase with particle diameter is found. When considering inter-particle collisions, however, the erosion ratio shows a non-linear increase,

for small particles at a lower rate than for large particles. The erosion growth rate for large particles seems to be constant again, at least for the considered size range. This nonlinear behaviour is

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

1099 1100 1101

APT 2152

No. of Pages 18, Model 5G

26 November 2018 16

S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

Fig. 15. Effect of particle mass loading ratio on bend erosion and comparison of two- and four-way coupled simulations; (a) penetration ratio in m/kg in the bend for glass beads particles; (b) averaged modulus of particle wall impact velocity; (c) average particle wall impact angle regarding the tangent plane to the wall at the particle impact point; (d) average particle wall impact kinetic energy along the bend outer wall for a 5 mm strip (size distribution 15–85 lm, D = 150 mm, bend radius 2.54
D, Uav = 27 m/s, Dc = 10°).

Fig. 16. Effect of particle mass loading ratio on bend erosion velocity and comparison of two- and four-way coupled simulations: penetration rate in m/s in the bend for glass beads particles along the bend outer wall for a 5 mm strip (size distribution 15–85 lm, D = 150 mm, bend radius 2.54
D, Uav = 27 m/s, Dc = 10°).

Fig. 17. Effect of particle mass loading ratio on bend erosion ratio [mm3/kg], closed line and left axis, and erosion rate [mm3/s], dashed line and right axis, comparing two- and four-way coupled simulations (erosion volume loss obtained for the entire bend inner surface (size distribution 15–85 lm, D = 150 mm, bend radius 2.54
D, Uav = 27 m/s, Dc = 10°).

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

APT 2152

No. of Pages 18, Model 5G

26 November 2018 S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

17

 A very important finding is that the consideration of a particle size distribution with the same number mean diameter yields much higher erosion rates compared to just taking monodisperse particles, which is very often done. This is mainly due to particles larger than the mean value being present in the size distribution, which have much higher impact energy.  Moreover, when particle mass loading is increased, the maximum penetration ratio is decreased due to the growing importance of inter-particle collisions and associated reduction in particle-wall impact velocity and impact angle. Finally, at higher mass loading, two-way coupling effects further reduce the averaged particle wall impact velocity in a horizontal to vertical bend.

1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156

Fig. 18. Effect of particle diameter (mono-sized) on bend erosion ratio [mm3/kg], comparing two- and four-way coupled simulations (erosion volume loss obtained for the entire bend inner surface, D = 150 mm, bend radius 2.54
D, Uav = 27 m/s, g = 0.3, Dc = 10°).

1110

the result of the particle behaviour with respect to inter-particle collisions, as discussed above. Similar to the wall collision rates presented in Fig. 8, but with an opposite trend, also at 30 lm there is a cross over in the specific erosion rate (Fig. 18). For particles smaller than this critical size of 30 lm inter-particle collisions increase erosion ratio and for larger particles again a decrease is observed which is growing with particle size due to the nonlinearity. The critical particle size is depending on operational conditions and the bend geometry.

1111

5. Conclusions

1112

In this paper, the combination of the Euler-Lagrange approach and the erosion model proposed by Oka et al. [3] has been applied to study the effect of wall roughness, inter-particle collisions and particle mass loading on the penetration ratio, as one measure of erosion severity, in an elbow flow configuration of a pneumatic conveying line. First of all, the applied erosion model was validated based on experimental data, yielding satisfactory agreement. It was however also demonstrated that wall surface structure, especially wall roughness, which was considered in the wall collision model, and also assumptions about the wall friction coefficient have a remarkable effect on erosion predictions. The major findings of this numerical study are:

1102 1103 1104 1105 1106 1107 1108 1109

1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142

 An increase in wall roughness decreases remarkably the penetration ratio and also changes the shape of the erosion pattern, eliminating the two antenna-like zones obtained for smooth wall computations. This is mainly associated with the stronger rebound of particle on rough walls and hence a global reduction of wall collision frequency. In addition the averaged modulus of the particle impact velocity is reduced and the averaged particle-wall impact angle increases.  Using higher wall friction coefficients in the wall collision model also yields a reduction of penetration ratio caused by a drop of the averaged modulus of the particle impact velocity.  On the other hand, penetration ratio reduces as a consequence of inter-particle collisions despite the particle-wall collision frequency is higher in four-way coupling associated with the trapping of particles within the dust rope. The reason for this reduction is found in a reduced modulus of particle impact velocity and a decrease in averaged impact angle by the trapping effect (i.e. particles are bouncing back and forth in the dense rope by inter-particle collisions).

The presented results reveal that a numerical calculation of particle induced erosion requires detailed knowledge about the necessary wall-collision model parameters, such as friction coefficient and restitution ratio. In addition wall roughness is extremely important to be considered. These phenomena strongly affect wall collision rates and hence predicted erosion rates. Therefore, actually also a modification of wall roughness structure during the erosion process has to be considered. Numerous calculations without considering inter-particle collisions were published in the past. However, even at moderate particle overall concentrations interparticle collisions will remarkably reduce erosion in a pipe bend caused by a kind of protection effect through the dust rope formed by inertia effects. Neglecting inter-particle collisions will therefore result in a remarkable overestimation of erosion rates.

1157

Acknowledgement

1171

The financial support of the Dirección de Investigaciones y Desarrollo Tecnológico of Universidad Autónoma de Occidente is gratefully acknowledged. S. Laín also thanks the MPS group of Faculty of Process and Systems Engineering, Otto-von-Guericke University (Germany) for their hospitality and support.

1172

References

1177

[1] S. Laín, M. Sommerfeld, Numerical calculation of pneumatic conveying in horizontal channels and pipes: detailed analysis of conveying behaviour, Int. J. Multiphase Flow 39 (2012) 105–120. [2] M. Sommerfeld, S. Laín, From elementary processes to the numerical prediction of industrial particle-laden flows, Multiphase Sci. Technol. 21 (2009) 123–140. [3] Y. Oka, K. Okamura, T. Yoshida, Practical estimation of erosion damage caused by solid particle impact: part 1: effects of impact parameters on a predictive equation, Wear 259 (2005) 95–101. [4] Q.H. Mazumder, S.A. Shirazi, B. McLaury, Experimental investigation of the location of maximum erosive wear damage in elbows, J. Pressure Vessel Technol. 130 (2008) 1–7. [5] M. Sommerfeld, S. Laín, Parameters influencing dilute-phase pneumatic conveying through pipe systems: a computational study by the Euler/ Lagrange approach, Can. J. Chem. Eng. 93 (2015) 1–17. [6] M.S. El Togby, E. Ng, M.A. Elbestawi, Finite element modeling of erosive wear, Int. J. Mach. Tools Manuf. 45 (2005) 1337–1346. [7] J. Bitter, A study of erosion phenomena, part I, Wear 6 (1963) 5–21. [8] I.M. Hutchings, R.E. Winter, Particle erosion of ductile metals: a mechanism of material removal, Wear 27 (1974) 121–128. [9] G.C. Pereira, F.J. de Souza, D.A. Martins, Numerical prediction of the erosion rate due to particles in elbows, Powder Technol. 261 (2014) 105–117. [10] Y. Zhang, E.P. Reuterfors, B.S. McLaury, S.A. Shirazi, E.R. Rybicki, Comparison of computed and measured particle velocities and erosion in water and air flows, Wear 263 (2007) 330–338. [11] X. Chen, B.S. McLaury, S. Shirazi, Application and experimental validation of a computational fluid dynamics (CFD) — based erosion prediction model in elbows and Plugged Tees, Comput. Fluids 33 (2004) 1251–1272. [12] I. Finnie, The Mechanism of Erosion of Ductile Metals, 3rd U.S. Nat. Congress of Applied Mechanics, ASME, New York, 1958, pp. 527–532. [13] A. Levy, Solid Particle Erosion and Erosion-Corrosion of Materials, ASM International, 1995. [14] I. Kleis, P. Kulu, Solid Particle Erosion Occurrence, Prediction and Control, Springer-Verlag, Heidelberg, 2008.

1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170

1173 1174 1175 1176

APT 2152

No. of Pages 18, Model 5G

26 November 2018 18 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268

S. Laín, M. Sommerfeld / Advanced Powder Technology xxx (xxxx) xxx

[15] M. Parsi, K. Najmi, F. Najafifard, S. Hassani, B.S. McLaury, S.A. Shirazi, A comprehensive review of solid particle erosion modeling for oil and gas wells and pipelines applications, J. Nat. Gas Sci. Eng. 21 (2014) 850–873. [16] A. Levy, P. Chik, The effect of erodent composition and shape on the erosion of steel, Wear 89 (1983) 151–162. [17] A. Levy, G. Hickey, Surface degradation of metals in simulated synthetic fuels plant environments, in: NACE Corrosion/82, International Corrosion Forum, 1982, p. 154. [18] Y. Oka, T. Yoshida, Practical estimation of erosion damage caused by solid particle impact: part 2: mechanical properties of materials directly associated with erosion damage, Wear 259 (2005) 102–109. [19] H.H. Uuemis, I.R. Kleis, A critical analysis of erosion problems which have been little studied, Wear 31 (1975) 359–371. [20] D. Mills, Erosive Wear Problems in Industry with Particular Reference to Process Plant, Power Station and Bulk Solids Handling Systems, SOLIDDEX’86 Conference/Exhibition, Harrogate, June, 1986. [21] X. Chen, Application of Computational Fluid Dynamic (CFD) to Single Phase and Multiphase Flow Simulation and Erosion Prediction (Ph.D. dissertation), Department of Mechanical Engineering, University of Tulsa, 2004. [22] T. Deng, A.R. Chaudhry, M. Patel, I. Hutchings, M.S.A. Bradley, Effect of particle concentration on erosion rate of mild steel bends in a pneumatic conveyor, Wear 258 (2005) 480–487. [23] C.A.R. Duarte, F.J. de Souza, V.F. dos Santos, Numerical investigation of mass loading effects on elbow erosion, Powder Technol. 283 (2015) 593–606. [24] M. Sommerfeld, S. Lain, Euler/Lagrange methods, in: E.E. Michalelides, C.T. Crowe, J.D. Schwarzkopf (Eds.), Multiphase Flow Handbook, second ed., CRC Press, Taylor & Francis Group, Boca Raton, 2017, pp. 202–242. [25] C.A.R. Duarte, F.J. de Souza, R.V. Salvo, V.F. dos Santos, The role of inter-particle collisions on elbow erosion, Int. J. Multiphase Flow 89 (2017) 1–22. [26] S.M. El-Behery, M.H. Hamed, M.A.E. -Kadi, K.A. Ibrahim, Numerical simulation and CFD-based correlation of erosion threshold gas velocity in pipe bends, CFD Lett. 2 (2010) 39–53. [27] X. Chen, B.S. McLaury, S. Shirazi, Numerical and experimental investigation of the relative erosion severity between plugged tees and elbows in dilute gas/solid two-phase flow, Wear 261 (2006) 715–729. [28] H. Hadzˇiahmetovic´, N. Hodzˇic´, D. Kahrimanovic´, E. Dzˇaferovic´, Computational fluid dynamics (CFD) based erosion prediction model in elbows, Tehnicˇki Vjesnik 21 (2014) 275–282. [29] A. Mansouri, H. Arabnejad, S.A. Shirazi, B.S. McLaury, A combined CFD/experimental methodology for erosion prediction, Wear 332–333 (2015) 1090–1097. [30] M. Sommerfeld, B. van Wachem, R. Oliemans, Best Practice Guidelines for Computational Fluid Dynamics of Dispersed Multiphase Flows, ERCOFTAC, Brussels, Belgium, 2008. [31] M. Sommerfeld, Numerical methods for dispersed multiphase flows, in: T. Bodnár, G.P. Galdi, Š. Neccˇasová (Eds.), Particles in Flows, Series Advances in Mathematical Fluid Mechanics, Springer International Publishing, Heidelberg, 2017, pp. 327–396. [32] M. Sommerfeld, Analysis of collision effects for turbulent gas-particle flow in a horizontal channel: Part I. Particle transport, Int. J. Multiphase Flow 29 (2003) 675–699. [33] M. Sommerfeld, Modellierung und numerische Berechnung von partikelbeladenen turbulenten Strömungen mit Hilfe des Euler/Lagrange Verfahrens. Habilitation Thesis, University Erlangen-Nürnberg, Shaker Verlag, Aachen, 1996. [34] S. Lain, M. Sommerfeld, Characterization of pneumatic conveying system using the Euler/Lagrange approach, Powder Technol. 235 (2013) 764–782.

[35] D.O. Njobuenwu, M. Fairweather, Modelling of pipe bend erosion by dilute particle suspensions, Comput. Chem. Eng. 42 (2012) 235–247. [36] N. Huber, M. Sommerfeld, Modelling and numerical calculation of dilute-phase pneumatic conveying in pipe systems, Powder Technol. 99 (1998) 90–101. [37] S. Laín, M. Sommerfeld, Euler/Lagrange computations of pneumatic conveying in a horizontal channel with different wall roughness, Powder Technol. 184 (2008) 76–88. [38] M.F. Göz, S. Laín, M. Sommerfeld, Study of the numerical instabilities in lagrangian tracking of bubbles and particles in two-phase flow, Comput. Chem. Eng. 28 (2004) 2727–2733. [39] M.F. Göz, M. Sommerfeld, S. Laín, Instabilities in Lagrangian tracking of bubbles and particles in two-phase flow, AIChE J. 52 (2006) 469–477. [40] M. Sommerfeld, G. Kohnen, M. Rüger, Some open questions and inconsistencies of Lagrangian dispersion models, in: Proc. 9th Symp. On Turbulent Shear Flows, Kyoto, Japan, 1993, paper 15-1. [41] M. Sommerfeld, N. Huber, Experimental analysis and modelling of particlewall collisions, Int. J. Multiphase Flow 25 (1999) 1457–1489. [42] S. Laín, M. Sommerfeld, J. Kussin, Experimental studies and modelling of fourway coupling in particle-laden horizontal channel flow, Int. J. Heat Fluid Flow 23 (2002) 647–656. [43] M. Sommerfeld, Validation of a stochastic Lagrangian modelling approach for inter-particle collisions in homogeneous isotropic turbulence, Int. J. Multiphase Flow 27 (2001) 1828–1858. [44] G. Kohnen, M. Rüger, M. Sommerfeld, Convergence behaviour for numerical calculations by the Euler/Lagrange method for strongly coupled phases, in: Crowe et al., (Ed.), Num. Meth. for Multiphase Flows, FED vol. 185, 1994, pp. 191–202. [45] I. Finnie, Erosion of surfaces by solid particles, Wear 3 (1960) 87–103. [46] H.C. Meng, Wear Modeling: Evaluation and Categorization of Wear Models, University of Michigan, Ann Arbor, MI, USA, 1994. [47] H.C. Meng, K.C. Ludema, Wear models and predictive equations—their form and content, Wear 181 (1995) 443–457. [48] C.B. Solnordal, C.Y. Wong, A. Zamberi, M. Jadid, Z. Johar, Determination of erosion rate characteristics for particles with size distributions in the low Stokes range, Wear 305 (2013) 205–215. [49] V.B. Nguyen, Q.B. Nguyen, Z.G. Liu, S. Wan, C.Y.H. Lim, Y. Zhang, A combined numerical-experimental study on the effect on the water-sand multiphase flow characteristics and the material erosion behaviour, Wear 319 (2014) 96– 109. [50] G. Grant, W. Tabakoff, Erosion prediction in turbomachinery resulting from environmental solid particles, J. Aircraft 12 (1975) 471–478. [51] C.B. Solnordal, C.Y. Wong, J. Boulanger, An experimental and numerical analysis of erosion caused by sand pneumatically conveyed through a standard pipe elbow, Wear 336–337 (2015) 43–57. [52] S. Laín, M. Sommerfeld, Influence of inter-particle collisions on erosion of pipe bends, ERCOFTAC Bull. 112 (2017) 10–16. [53] F.A. Bikbaev, V.I. Krasnov, V.L. Berezin, I.B. Zhilinski, N.T. Otroshko, Main factors affecting gas abrasive wear of elbows in pneumatic conveying pipes, Chem. Pet. Eng. 9 (1973) 73–75. [54] S. Laín, M. García, B. Quintero, S. Orrego, CFD numerical simulations of Francis turbines, Rev. Facultad de Ingeniería Universidad de Antioquia 51 (2010) 24– 33. [55] A.D. Caballero, S. Laín, Numerical simulation of non-Newtonian blood flow dynamics in human thoracic aorta, Comput. Methods Biomech. Biomed. Eng. 18 (2015) 1200–1216.

Please cite this article as: S. Laín and M. Sommerfeld, Numerical prediction of particle erosion of pipe bends, Advanced Powder Technology, https://doi.org/ 10.1016/j.apt.2018.11.014

1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324