Particle-Laden Flows: Theory and Engineering Applications: The multiscale erosion model

Erosive wear of prototype-scale pelton turbines

The multiscale model is composed of two sequentially coupled submodels. In the microscale model, the sediment impacts and the resulting material damage and removal are considered. The mass and linear momentum conservation laws are solved for the sediments and the base material, referred to hereafter as the solid. Both materials are modeled as homogeneous and isotropic, together with the temperature-corrected Mie-Grüneisen equation of state to close the system. Whereas the sediments are modeled as elastic, the solid is modeled as elastoplastic by means of the Johnson-Cook strength and damage models, which define the yield stress and failure plastic strain as functions of the strain, strain-rate, temperature and triaxiality states. The material frictional and thermoplastic heating are also taken into account. In the macroscale model, the turbulent sediment transport, the impact detection between wall and sediments, and the accumulation of eroded mass are calculated on the domain of interest. The mass and linear momentum conservation laws are solved for the fluid, which is modeled as Newtonian and weakly compressible. The Tait equation of state and the standard k-ε turbulence model with wall function are used for closure. The sediments, considered as discrete masses, are tracked through the domain using a one-way coupling scheme. The hydrodynamic force on the sediments considers the effect of drag, added mass and pressure gradient. The important effect of turbulence on the sediment dispersion is considered via a continuous random walk model based on the stochastic Langevin equation.

Microscale simulation of sharp sediments repeatedly impacting the solid specimen (left); the solid is colored according to the Johnson-Cook model damage parameter, whereas the sediments are colored according to the magnitude of their velocity. Macroscale simulation of the turbulent flow exiting a Pelton bucket (right); the fluid is colored according to its velocity magnitude, whereas the sediments are too small to be visible. (Leguizamón et al., IJTPP, 2019).

The models are coupled in a sequential manner. First, the space of possible impact conditions is explored by independent microscale simulations, each of which involves 50-100 impacts at constant angle and velocity on a microscopic solid specimen. These simulations render the average restitution coefficients and the steady-state erosion ratio for each impact condition. Second, the macroscale sediment transport simulation is performed; every time an impact against the wall is detected, the microscale simulation results are interpolated in order to determine the sediment rebound velocity, using the restitution coefficients, and the amount of mass eroded, using the steady-state erosion ratio and the sediment mass. After enough impacts have occurred, converged distributions of eroded mass, erodent mass, impact angle and impact velocity are obtained on the surface of interest.

Example simulation results and model validation
Multiscale simulations of the erosion of a static Pelton bucket have been recently performed. Some of the results of the simulation are illustrated further down, including the distributions of average impact angle, average impact velocity, erodent mass and eroded mass. The impact angle is highest at the splitter, and fairly uniform elsewhere, whereas the impact velocity is lowest at the deepest part of the bucket and at the splitter, and increases towards the outlet. The erodent flux is highest where the surface curvature is highest, as expected: in the splitter and at the deepest part of the bucket. The aforementioned distributions allow understanding the distribution of eroded mass, which is a direct consequence of where the sediments tend to impact, and at what angle and velocity. The eroded mass is greatest at the splitter and increases gradually towards the bucket outlet.

Distributions of average impact angle, average impact velocity, impacted erodent mass and eroded mass on the surface of a Pelton bucket. (Leguizamón et al., IJTPP, 2019).

The eroded mass distribution can be used to compute the erosion depth distribution, which in turn can be used to validate the model by comparison with the corresponding experimental data, as illustrated further down. Although the simulation results are noisy because of the relatively low number of sediment impacts, the erosion depth is predicted with considerable accuracy. However, note that the comparison is not completely justified because of several discrepancies between the simulation setup and the experimental field study, including the bucket geometry.

Erosion depth distribution along the pitch diameter position of a Pelton bucket after 3180 hours of operation; experimental data by Rai et al., Wear, 2017. (Leguizamón et al., IJTPP, 2019).

The erosion depth validation results of a different test case, namely one in which the simulated runner is rotating and is geometrically identical to the one investigated in the field study, are presented further down. Yet again, the quantitative agreement between the multiscale simulation and the corresponding experiment is remarkable. An average error of about 30% has been calculated for this realistic test case, which involved the erosive wear of a 84 MW prototype-scale Pelton runner along 21 months of operation.

Erosion depth ed, normalized by the corresponding average experimental measurements ed, exp, at four points along the deepest bucket section (left), four points along the bucket outlet (center), and average value along four sections (right). The red lines, lower and upper box bounds, and whiskers represent the median, 25th and 75th percentiles, and extreme values, respectively. (Leguizamón et al., submitted to Renewable Energy, 2019.)