A hybrid DEM CFD approach for solid-liquid flows

A hybrid DEM CFD approach for solid-liquid flows

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2014,26(1):19-25

DOI: 10.1016/S1001-6058(14)60003-2

A hybrid DEM/CFD approach for solid-liquid flows*

QIU Liu-chao ???????

College of Water Resources and Civil Engineering, China Agricultural University, Beijing 100083, China, E-mail: qiuliuchao@cau.edu.cn WU Chuan-yu

Department of Chemical and Process Engineering, University of Surrey, Guildford, Surrey, GU2 7XH, UK (Received May 24, 2013, Revised September 1, 2013)

Abstract: A hybrid scheme coupling the discrete element method (DEM) with the computational fluid dynamics (CFD) is developed to model solid-liquid flows. Instead of solving the pressure Poisson equation, we use the compressible volume-averaged continuity and momentum equations with an isothermal stiff equation of state for the liquid phase in our CFD scheme. The motion of the solidphase is obtained by using the DEM, in which the particle-particle and particle-wall interactions are modelled by using the theoretical contact mechanics. The two phases are coupled through the Newton’s third law of motion.To verify the proposed method, the sedi- mentation of a single spherical particle is simulated in water, and the results are compared with experimental results reported in the literature. In addition, the drafting, kissing, and tumbling (DKT) phenomenon between two particles in a liquid is modelled and rea- sonable results are obtained. Finally, the numerical simulation of the density-driven segregation of a binary particulate suspension in- volving 10 000 particles in a closed container is conducted to show that the presented method is potentially powerful to simulate real particulate flows with large number of moving particles.

Key words: discrete element method (DEM), computational fluid dynamics (CFD), solid-liquid flows, sedimentation, two-phase flow, numerical simulation

Introduction??

The solid-liquid flows can be found in many pro- cesses in chemical, petroleum, agricultural, biochemi- cal and food industries. To improve the design of the process equipment while avoiding tedious and time consuming experiments, numerical approaches were widely employed during the last decades and a num- ber of numerical models were developed. For example, Rong and Zhan[1] used the DEM-CFD approach for modelling spouted beds, Wan and Turek[2] applied the multigrid fictitious boundary method (MFBM) to simulate solid-liquid flows, Tong et al.[3] used a DEM- CFD method to simulate the powder dispersion in a cyclonic flow, Cate et al.[4] simulated a single sphere settling under gravity using the lattice-Boltzmann me-

*Project supported by the National Natural Science Foun- dation of China (Grant No. 11172321).

Biography: QIU Liu-chao (1971-), Male, Ph. D., Associate Professor

thod (LBM), Chen et al.[5] applied a dissipative parti- cle dynamics (DPD) for simulating a single sphere se- ttling in a square tube, Yang et al.[6] adopted the dis- crete particle method for analysing the sedimentation of microparticles, Potapov et al.[7] and Qiu[8] modelled the liquid-solid flows using a coupled smoothed parti- cle hydrodynamics (SPH) and DEM. Over the last two decades, a coupled DEM/CFD was developed and ad- vanced for modelling the fluid-solid flows, and for ef- ficiently analyzing the gas-solid flows[9,10]. In addition, detailed information at the micro-scale can be obtai- ned with the DEM/CFD.

However, most DEM/CFD methods were develo- ped for compressible fluids (i.e., gases) based upon the ideal gas law, which cannot be used for the parti- cle motion in liquids that are generally incompressible. The liquid flow is conventionally simulated by using the incompressible Navier-Stokes equations and the pressure is obtained by solving a Poisson equation. Solving the pressure Poisson equation is often the most costly step in these schemes, to hinder their app- lications in analyzing particulate systems. An alterna- tive approach for numerical modelling of particulate

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systems involving a nearly incompressible Newtonian fluid is proposed in this study, which is stimulated by the weakly compressible flow method developed by Brujan[11] as the fluid solver. Instead of solving the pressure Poisson equation, the compressible volume- averaged continuity and momentum equations are used, but with an isothermal stiff equation of state for the fluid phase.

This paper is organized as follows. The numeri- cal methods are first discussed, followed by the model validation. Then the numerical simulation of the den- sity-driven segregation of a binary particulate suspe- nsion in a closed container is presented. The conclu- sions and discussions are at the end.

1. The CFD/DEM method

In order to model incompressible fluids in the DEM/CFD, an equation of state for weakly compre- ssible fluids is adapted in place of the ideal gas law. This is incorporated into the DEM/CFD model origi- nally developed by Kafui et al.[9], with the original al- gorithms kept intact to retain its capability for gas- solid flows. For completeness, the theoretical aspect of the modified DEM/CFD is presented in this section. 1.1 Governing equations for liquids

The compressible volume-averaged Navier- Stokes equations of continuity and momentum with an additional stiff equation of state are used for mode- lling liquid flows. The volume-averaged compressible equations of continuity and momentum conservation for fluid motions are[9]:w(HUf)wt

+??(HUfu)=0

(1)

w(HUfu)wt

+??(HUfuu)=????p+??Wf??1

'V|nc

f

fpi

+HUfg

(2)

c

i=1

whereu,UfandHare the velocity, the density and the volume fraction of the fluid, respectively,gis the gravity acceleration, Wf is the fluid viscous stress, evaluated as

W2f=[(Pb??3

Ps)????u]G+Ps[(??u)+(??u)T]

whereG is the Kroneker delta, Pb and?Ps are

the bulk viscosity and the shear viscosity, respectively. ffpi is the fluid-particle interaction force acting on

particlei,ncis the number of particles in a fluid cell of volume 'Vc,ffpi is given as

ffpi=??vpi????p+vpi????Wf+Hfdi

(3)

wherevpiis the volume of particlei, andfdiis the drag force in the direction of the relative velocity bet- ween fluid and particle. In the above equations, the fluid volume fraction H is obtained from the relation

nc

v

pi

H=1??

|i=1

'V (4)

c

Since fdi is a quantity usually obtained from experi- mental correlations, the specific form of the drag force depends on the correlations used. In this study, Di Felice’s[12] correlation is employed in evaluating the

drag force fdi on

particle if1Sd2

pi??(F+1)

di=2CDiUf4

H2juj??vi(uj??vi)Hj

(5)

where the subscript j for the fluid velocity and the void fraction represents the computational fluid cell in which the particle i resides and CDi, the fluid drag coefficient for a single, unhindered particle, is evalua- ted by using

2

C=§¨¨0.63+4.8·

DiRe0.5?

(6)

?pi?1

The term H??(F+1)

j

in Eq.(5) is a correction due to the presence of other particles, and the correlation equa-

tion obtained for F by Di Felice[12]

??0.65expa?(1.5??lg2F=3.710Repi)o

???2??

(7)

??

takes account of the variation of this exponent in the intermediate flow regime as well as the near constant values in the low and high Reynolds number flow re- gimes. The particle Reynolds number Repi, based on the superficial slip velocity between fluid and particle, is

RedpiHjuj??vi

pi=

UfP (8)

s

To close the system, an equation of state (EOS) relating the density to the pressure is required. For the liquid-solid flow systems to be considered in this study, a pressure-density relationship of the modified Tait equation of state[11] is incorporated, i.e.,

p+Bn

p0+B=§¨Uf·

¨?U0??

(9)

f1

whereBandnare constants,pis the fluid pressure, p0andU0fdenote the undisturbed fluid pressure and

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