Comprehensive Euler/Lagrange modelling including particle erosion for confined gas-solid flows

The present research aims to assess the capability of a comprehensive Euler/Lagrange approach for predicting gas-solid flows and the associated solid particle erosion.The open-source code OpenFOAM®4.1 was used to carry out the numerical simulations,where the standard Lagrangian libraries were substantially extended to account for all necessary models.Particles are tracked considering both translational and rotational motion as well as all relevant forces,such as gravity/buoyancy,drag and transverse lift due to shear and particle rotation.The tracking time step was dynamically adapted ac-cording to the locally relevant time scales,which drastically reduces computational times.Stochastic approaches are adopted to model particle turbulent dispersion,particle collisions with rough walls and particle-particle interactions.Five solid particle erosion models,available in the literature,were considered to estimate pipe bend erosion.Three study cases are provided to validate the adopted nu-merical approach and erosion models extensively.The first case intends to evaluate the ability of the extended CFD code to predict the behaviour of gas-solid flows in pneumatic conveying systems.This goal is achieved by comparing the numerical results with the experimental data obtained by Huber(1997)and Huber and Sommerfeld(1994,1998)in a pneumatic conveying system.Here,the importance of considering inter-particle collisions and surface roughness for predicting particle velocity,mass flux and mean diameter distributions in gas-solid flows is highlighted.The second and third case intend to evaluate the ability of the erosion models in estimating bend erosion in diluted gas-solid flows.The erosion data obtained experimentally by Mazumder et al.(2008)and Solnordal et al.(2015)in very dilut pneumatic conveying systems is used for validating the numerical results,neglecting now inter-particle collisions and two-way coupling.Besides a comprehensive analysis of the different influential properties on erosion,the innovation of the present study is as follows.For the first time also a temporal modifi-cation of the surface roughness due to the erosion was considered in the simulations obtained from previous measurements(Novelletto Ricardo&Sommerfeld,2020).As the surface roughness is increased due to erosion,eventually erosion rate becomes lower.This is the result of diminishing wall collision frequency.Simulations for several degrees of surface roughness showed that larger roughness is coupled with a drastic reduction of erosion.Hence,numerical simulations neglecting wall surface roughness are not realistic.The consideration of a particle size distribution instead of mono-sized computations showed a possible reduction of erosion rate.The detailed analysis of the different single-particle erosion models revealed that the model proposed by Oka et al.(2005)and Oka and Yoshida(2005)yields the best agreement with the measurements,however particle and wall properties are needed.

Efficient numerical algorithms for the phase field crystal equation

In this paper,based on the Lagrange Multiplier approach in time and the Fourierspectral scheme for space,we propose efficient numerical algorithms to solve the phase field crystal equation.The numerical schemes are unconditionally energy stable based on the original energy and do not need the lower bound hypothesis of the nonlinear free energy potential.The unconditional energy stability of the three semi-discrete schemes is proven.Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of our proposed schemes.