*2.2.2 Eulerian-Lagrangian approach*

The Eulerian-Lagrangian formulation allows the trajectories of particles within the discrete phase to be tracked. As before, the carrier fluid (continuous phase) is modelled following the Eulerian framework. To model the erosion process in turbulent flows, steady state Reynolds-averaged Navier-Stokes (RANS) equations have been combined with a variety of turbulence models to simulate fluid flows [32]. This approach has been used somewhat successfully over the past couple of decades [2, 33]. While the RANS approach can be used to establish the timeaveraged erosion characteristics of a particle-laden fluid, it is unable to establish the unsteady nature of turbulent flows being modelled. To simulate unsteady properties of pipe flows such as the evolution of vortices and recirculation, large eddy simulation (LES) algorithms have been developed. A detailed description of LES can be found in [32]. The performance of both algorithms has been evaluated. Wang et al. [34] studied erosion in pipe elbows using an LES Euler-Lagrange model. Results obtained using the RANS and LES were compared. Results were verified using experimental results taken from [35]. Good agreement between velocity profiles predicted by the LES method and experimental values was noted. In comparison, the RANS method was found to overpredict the fluid velocity in the near-wall

region. The increased accuracy of the LES model was attributed to its ability to model the unsteady secondary flow at the inner radius.

The Lagrangian framework resolves the motion of particles based on a translational force balance for each entity. The Discrete Phase Model (DPM) tracks the movement of particles through the continuous phase. In the DPM each modelled entity represents a 'parcel' of particles having similar dynamic properties [36]. Another approach to particle tracking is the Discrete Element Method (DEM). Unlike the DPM, individual particles are tracked and their equations of motion solved. The benefit of this method is that information related to inter-particle collisions can be calculated using the 'soft sphere' approach, making the DEM suitable for flows with high solid concentrations. The applicability of the DPM is limited to solid concentrations where the influence of inter-particle collisions can be neglected. Wang et al. [2] found the influence of inter-particle collisions to be negligible at solid concentrations up to 5.37% wt., while Kloss et al. [37] limited their use of DPM modelling to below 5% by volume. Lagrangian inter-particle collision models have been used in [38, 39] to extend the range of applicability to higher concentrations of solid particles. However, their use remains limited due to the increased computational burden and the number of difficult-to-estimate parameters these models rely on [40].

Coupling between the DEM and DPM has been proposed to improve the accuracy of the CFD-DPM approach without the full computational expense of the CFD-DEM method. Kloss et al. [37] used the CFD-DPM-DEM approach to model the dispersed phase in air-particle flows. They showed that the simulation speed of various systems using DEM could be significantly increased by switching to the DPM methodology in regions of low particle concentration without a significant effect on the results of the simulation. By extending this technique to solid-liquid flows the ability of CFD packages to model regions of high concentration could be improved.

#### **2.3 Advancements in CFD**

#### *2.3.1 Flow modelling*

It is widely accepted that erosion is a time-dependent process. The formation of wear scars takes time to develop and is most severe in areas with highly disturbed flows (such as in elbows, contractions, and tees). It is known that deformations to wall geometry can modify the flow field and cause changes to the local erosion rate. Despite this, most CFD packages assume that cavities formed are insignificant to the flow field. This assumption can lead to significant errors in erosion field calculations, especially in areas with sharp geometrical changes. López [41], Dong [42] and Agrawal et al. [43] all proposed modifications to CFD solvers to dynamically change wall geometry based on the local erosion rate.

A López et al. [41] utilised open source CFD software OpenFOAM. Surface erosion was calculated using built-in functions to establish an erosion vector for each point on the surface. This material loss was used to calculate the change in the shape of the solid surface. The geometry of the solid surface was modified by moving points of the mesh depending on the material loss magnitude. This process was repeated at regular intervals, with the flow field being re-calculated at each instance.

Using commercial CFD software Ansys Fluent, Dong et al. [42] proposed a methodology using a dynamically deforming mesh to model the deformed surface profile due to material erosion. Material removal was converted into a wall deformation using a user-defined function (UDF) within Fluent. As in [41], the mesh was updated

#### *Erosion-Corrosion in Pipe Flows of Particle-Laden Liquids DOI: http://dx.doi.org/10.5772/intechopen.107231*

at regular time intervals. The flow field was updated according to the new deformed wall shape. Their methodology was implemented in the erosion modelling of an economizer bank used in coal-fired power plants. A good agreement between the erosion profile predicted using the CFD model and the on-site test sample was noted. By utilising the CFD model, the evolution of erosion profile and particle trajectories could be seen. While their results appeared promising, no comparison to CFD models without the moving mesh approach was made, meaning the accuracy increase with their methodology cannot be quantified.

Similarly, Agrawal et al. [43] developed a moving-deforming-mesh (MDM) model within CFD software Ansys Fluent. The mesh deformation was calculated using UDFs within the software. Based on the local erosion rate, the geometry of the solid wall and its computational mesh were updated at regular time intervals. The flow field was recalculated following each update. It was found that the dynamically changing flow fields were able to capture the development of flow features which were not present in the undeformed model. Their approach proved capable of predicting secondary erosion features not present in non-MDM CFD models. Agrawal concluded that the effect of the MDM was most pronounced at sharp geometrical features (such as changes in section or elbows) where erosion caused significant changes to the flow field. A case study of a mitre-bend was used to illustrate the differences between models with and without the MDM algorithm.

Additional challenges in CFD modelling come when trying to calculate particle motion near solid boundaries. For erosion estimates to be accurate, the motion of the particle must be equivalent to that found in real-world scenarios. Different researchers have tried to investigate near wall effects and rebound characteristics of the discrete phase. Issues in particle motion and rebound modelling were demonstrated by Karimi et al. [31]. From their review of literature, it was found that CFD packages over-estimated the true rate of erosion. They found that modelling errors led to small particles becoming trapped in eddies in the near wall region, leading to multiple wall impacts. This non-physical behaviour was thought to contribute to some of the overestimates commonly seen in CFD modelling. It was found that more realistic erosion estimates could be obtained by limiting the number of impacts a given particle to one. It was concluded that current rebound models are not simulating small particle behaviour after wall impacts correctly. More work is therefore required in this area.

#### *2.3.2 Wall erosion modelling*

Accurately turning wall impingements into an estimate of material loss is perhaps the biggest challenge in CFD erosion modelling. The process is highly complex, and research is still ongoing to understand the exact mechanisms behind erosion. Since the 1990s computational methods have been used to simulate erosive wear [44]. Erosion modelling using the Finite Element Method (FEM) has been successfully implemented to produce fairly accurate results [44–47]. The advantage of FEM is that erosion is calculated directly from material properties and the underlying physics. The drawback of this method is that it is limited to domain sizes at the same scale of the particles and for only a handful of collisions. This limits its use in CFD packages where both the scale and number of collisions are orders of magnitude higher.

The approach taken by CFD modellers is to disregard the physics at the microscale, instead focusing on the macroscale model encompassing the domain of interest. Trajectories and velocities of particles are calculated. Material removal due to wall impingements is estimated using empirical relationships derived from jet impact test

results. While popular due to the ease at which such relations can be incorporated into CFD packages, the validity of the results is limited. Empirical relations fail to account for the variation of particle size in naturally occurring materials and the non-linearity of the erosion process. Jet impact tests are carried out at higher velocities than most fluid flows, meaning that relationships drawn are often outside of the experiment's calibration range. Also, as was noted in the introduction, relations derived from experiments vary massively depending on methodology and test conditions. Even with careful application, it is not certain that test results are representative of the environment to be modelled.

Leguizamón et al. [47] proposed a multi-scale approach to surface erosion modelling. They created a database of physically determined micro-scale erosion results, calculated using the FEM approach. Particle collisions were modelled under a variety of conditions. These results were used to calculate material loss due to collisions within a macroscale model. The main benefit of this method is that erosion rates are based on detailed impact simulations as opposed to experimentally determined erosion correlations. To ensure the accuracy of this method a reliable and comprehensive set of erosion tests were carried out at the microscale. Validation results proved encouraging with significant improvements in accuracy over correlation-based approaches. Unfortunately, the proposed approach comes at a massive computational expense meaning that the method is unfeasible for full-scale modelling.

An alternative approach to improve the accuracy of wall material removal models was proposed by Mansouri [48] and later modified by Messa et al. [40]. Their solution consisted of an upgraded CFD/experimental methodology to calibrate coefficients used in the empirical erosion models based on slurry jet impingement tests. The slurry jet tests were performed on aluminium and curved glass reinforced epoxy samples. One-way coupling was used in the simulation owing to the relatively low volume fraction of solid. This was done to reduce the computation time as the trajectories of fluid parcels could be decoupled from the solution of the carrier fluid flow. The CFD simulation was carried out using ANSYS FLUENT and particle trajectories were calculated using the Lagrangian method. Modifications to the CFD code were implemented to account for the influence of particle shape on particle-wall impingement characteristics. Additionally, to reduce uncertainty due to the point-particle approximation, impact characteristics were evaluated at half of the particle diameter from the wall. This methodology was originally proposed by Messa and Wang [49], who found that a region of high drag near the wall resulted in an underestimation of impact velocity. An improvement in the reliability of CFD-based erosion predictions was noted. It was concluded that the method represented a practical compromise in the prediction of erosion using empirical models without the need for modelling at the micro-scale.

Wee et al. [11] sought to improve erosion damage prediction of CFD approaches by implementing the Rosin Rammler particle size distribution model. Simulations were carried out using Ansys Fluent 17.2. Two-phase flow and particle wall interactions were solved using the Euler-Lagrange model. Simulations were carried out using both water-sand and air-sand flows, to compare the influence of carrier-fluid on erosion rate. Results from the CFD simulation were compared with experimental results. Implementation of the Rosin Rammler particle size model reduced the error in CFD results from between 6.81–24.31% and below 5% for all test cases. Erosion wear was found to be up to 97.44% lower when using water as the carrier fluid. This is because sand particles follow the streamlines in the water much better than air, as characterised by their respective Stokes numbers.

### **2.4 Studies using CFD**

To reduce costs associated with erosion damage in sections of pipeline, many have turned their attention to optimising pipe geometry. Changes in cross-section, curvature and internal profile have all been suggested as methods of reducing E-C [50]. Internal ribs, a vortex chamber and twisting sections have been found to reduce erosion in elbows by modifying flow conditions within the pipe. However, in all these studies only gas-solid mixtures were considered. Li et al. [50] used a CFD-DEM simulation to model the influence of wall shape on wear rates in the vicinity of a pipe bend. The working environment consisted of a two-phase flow with particle concentrations of 1–10% and 1–3 mm in diameter. By adding a solid protrusion to the internal surface of the bends outer radius at the region at which particles first contact the wall it was found that the wear rate could be reduced significantly. The 'bump' reduced the kinetic energy of particles entrained in the flow and the number of times particles collide with the wall. The effectiveness of the bump in reducing wall wear was subject to other factors, including particle size, flow velocity and mass flow rate of particles.

Okhovat et al. [51] used a combined mathematical and CFD approach to model erosion-corrosion in straight and contracted pipe sections. COMSOL Multiphysics v3.5a was used to model fluid flow, while corrosion rate was determined as a function of oxygen concentration in the near wall regions. The pipe geometry used was a converging-diverging section, with an inlet diameter of 38.6 mm, contraction diameter of 21.2 mm and outlet diameter of 42.5 mm. When compared with experimental results, erosion estimates using the CFD model offered reasonably good agreements in the inlet and contracted region, but significantly underpredicted E-C in the expanded (downstream) section. This was put down to the momentum model used. Although the effects of both erosion and corrosion were predicted, their synergism was assumed negligible, which would likely have led to an underprediction of the true material loss rate.

It is not uncommon for the results of CFD simulations to be combined with experimental work. Many have used this approach to verify results from experimental tests. For example, More et al. [52] investigated the influence of impact angle on erosion wear in mild steel pipelines handling coal ash slurry. Findings from CFD simulations were verified against slurry pot test results. Impact angles ranged between 7 and 90°. All tests were carried out at slurry velocities of 4 m/s. While similar trends between impact angle and erosion rate were noted, CFD simulations were found to underpredict erosion rate by an average of 21%. The author suggested that difficulties exactly replicating the nature of coal ash particles in the CFD simulation may have attributed to this error. An underprediction of impingement velocity, or miscalculation of flow regime could have also attributed to this error. Speculatively, it could in part be due to inaccuracies in fluid models used by the CFD solver.

CFD has been used to model the influence of particle size on erosion rate. As part of his PhD thesis, Braut [36] studied the influence of nano-sized particles on erosion. CFD studies were carried out using commercial software STAR-CCM+ and verified using experimental data. Particle size varied between 1 and 500 μm. Erosion rate was found to increase with particle diameter, despite a larger number of particles at smaller diameters. This was put down to higher kinetic energies of larger particles. Erosion pattern was also found to change with particle size. It was suggested that this was because smaller particles were more affected by the fluid flow, as they had lower inertia.

Li et al. [53] used combined results from CFD and experimental testing to establish the effect of large particles on pipe erosion. The particles used were 3 mm in diameter at mass concentrations from 1 to 15%. An aluminium sheet was fixed to the inside of the test bend to establish the erosion profile. The resulting profile was found to be corrugated, a result of the 'bouncing' of the particles along the bends outer radius. The size of this 'erosion ripple' was found to be dominated by the concentration of solid particles. The observed periodic wave pattern was found to be consistent with that predicted by the numerical model.

Using a submerged impingement jet, Owen et al. [21] investigated the influence of impingement conditions on erosion-corrosion of X65 carbon steel. Particle trajectories within the jet were predicted using COMSOL Multiphysics CFD software. Impact angles and velocities of particles in the jet were predicted. Within the impingement zone, sites with differing degradation mechanisms were identified. The synergy between erosion and corrosion was investigated. Under these test conditions (60°C, pH 4.7 2% NaCl solution, 1 g/L sand particles at 20 m/s) corrosionenhanced erosion was found to be much more significant than erosion-enhanced corrosion.
