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Drag models for the spherocylinder CFDEM model

In this document, ~u is used to denote the relative velocity between particle and fluid.

~u = ~uf − ~up

Holzer and Sommerfeld, Di Felice

The drag force on a single particle is given by

1 π

F~D0 = ~u|~u|CD ρ d2s

2 4

Where the drag coefficient is given by Holzer and Sommerfeld [1]:

CD =

16 1

3 1

8 1

0.4(− ln φ)0.2 1

p +

√ +√

3 + 0.4210

Re φk

Re φ

φ⊥

Re φ 4

Where

Re =

|~u|ds

ν

φ⊥ =

As,⊥

Ap,⊥

φk =

2As,k

Ap,tot − 2Ap,k

The drag force on a particle in presence of other particles is given according to Di Felice [2]:

F~D = F~D0 2−β

(1.5 − ln Re)2

β = 3.7 − 0.65 exp −

2

Ergun Equation

The pressure drop over a CFD cell is given by the Ergun equation [3]

~ 0 (1 − )2

~ 0 |U

~ 0 (1 − )

∆P

µU

ρ|U

= 150

+

1.75

L

(φds )2 3

φds 3

The total force on all particles in a cell is given by

Ftot,cell =

∆P

Vcell

L

The force on each particle is given by [4]

Fpart =

Ftot,cell

Vpart

∆P Vpart

= Ftot,cell

=

npart

Vcell (1 − )

L(1 − )

~ 0 (1 − )

~ 0 |U

~0

µU

Fpart

ρ|U

= 150

+

1.75

Vpart

(φds )2 2

φds 2

~ 0 we substitute ~u.

For µ we substitute νρ, for U

~uVpart ρ

ν(1 − )

F~part =

150

+ 1.75|u|

φds

φds

1

In order to decide which drag model to use, the minimum of the Holzer & Sommerfeld / Di Felice

and the Ergun drag is taken. This ensures that the Ergun drag is used in dense regions.

F~D = min(F~D,H&Z/DF , F~D,Ergun )

References

˜

1. HA¶lzer,

A., & Sommerfeld, M. (2008). New simple correlation formula for the drag coefficient of

non-spherical particles. Powder Technology, 184(3)

2. Di Felice, R. (1994). The voidage function for fluid-particle interaction systems. International

Journal of Multiphase Flow, 20(I)

3. Ergun, S. (1952). Fluid flow through packed columns. Chem. Eng. Prog., 48

4. Gidaspow, D. (1994). Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. London: Academic Press Inc.

2

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