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A lattice Boltzmann method for simulating transport and agglomeration of resolved particles

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Abstract

The main purpose of this contribution is to present a lattice Boltzmann method for modelling the transport, collision and agglomeration of freely moving spherical particles and agglomerates. In order to take the hydrodynamic interaction between fluid and particles into account, the particle surface is fully resolved by the numerical grid using a curved no-slip boundary condition. In addition to various test cases with sedimenting single particles and particle pairs, a comparison with a finite element simulation is performed to evaluate the LBM-based treatment of flow-induced particle forces for gap widths smaller than the resolution limit of the fluid. Furthermore, the influence of viscous forces on the motion of approaching particles is analysed. As a final step, first results on the transient agglomeration inside a poly-sized particle cluster settling under gravity are presented for demonstrating the applicability of the code to more complex problems such as agglomeration in turbulent two-phase flows. The obtained agglomerate morphologies are characterised by various structural parameters such as a convex hull-based porosity and the radius of gyration. In the simulations, the observed particle Reynolds numbers are in the range [0.2, 84.8].

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Abbreviations

BGK:

Bhatnagar–Gross–Krook

DKT:

Drafting–kissing–tumbling

FDM:

Fictitious domain method

FEM:

Finite element method

LBM:

Lattice Boltzmann method

IBM:

Immersed boundary method

ODE:

Ordinary differential equation

PIV:

Particle imaging velocimetry

A VES :

Surface area of the volume equivalent sphere (m2)

A Agg :

Agglomerate surface area (m2)

c :

Grid constant (m/s)

c D :

Drag coefficient (–)

d P :

Particle diameter (m)

\({\bar{{d}}_P}\) :

Mean primary particle diameter (m)

F :

Force acting on a particle (N)

F Stokes :

Stokes drag force (N)

F ext :

External volume force (N)

F σi :

Discrete force at the particle surface (N)

f :

Distribution function (kg s3/m6)

f eq :

Equilibrium distribution (kg s3/m6)

f 0 :

Oscillation frequency (1/s)

f σ i :

Discrete distribution function (kg/m3)

\({f_{{\sigma} i}^{\ast} }\) :

Post-relaxation discrete distribution function (kg/m3)

\({f_{{\sigma}\, i}^{{\ast}{\ast}} }\) :

Post-propagation discrete distribution function (kg/m3)

\({f_{{\sigma}\, i}^{\rm eq} }\) :

Discrete equilibrium distribution (kg/m3)

\({f_{{\sigma} \, i}^{\rm neq} }\) :

Discrete non-equilibrium distribution (kg/m3)

G :

Fluid shear rate (1/s)

g :

Gravitational acceleration (m/s2)

h :

Gap height (m)

I :

Particle momentum (N s)

J P :

Particle moment of inertia (kg m2)

m P :

Particle mass (kg)

n PP :

Number of agglomerated primary particles (−)

q σ i :

Discrete relative distance (−)

R g :

Radius of gyration (m)

Re P :

Particle Reynolds number (−)

Sr P :

Particle Strouhal number (−)

St :

Particle Stokes number (−)

T :

Oscillation period (s)

T :

Torque acting on a particle (N m)

T σi :

Discrete torque at the particle surface (N m)

t :

Time (s)

U 0 :

Oscillation base velocity (m/s)

u :

Fluid velocity (m/s)

\({\tilde{{\bf u}}}\) :

Extrapolated fluid velocity (m/s)

u G :

Shear rate-based fluid velocity (m/s)

u P :

Particle velocity (m/s)

u Pij :

Modulus of the relative particle velocity (m/s)

u P,∞ :

Particle sedimentation velocity in an infinite medium (m/s)

u * P :

Post-collision particle velocity (m/s)

u S :

Velocity at the solid node (m/s)

V D :

Displaced volume (m3)

V H :

Volume of the convex hull (m3)

V P :

Particle volume (m3)

V S :

Solid-space within the convex hull (m3)

V V :

Void space within the convex hull (m3)

X 0 :

Oscillation amplitude (m)

x :

Position (m)

x F :

Position fluid node (m)

x FF :

Position of the next neighboured fluid node (m)

x P :

Particle position (m)

x Pij :

Modulus of the relative particle position (m)

x P,COM :

Position of particles centre of mass (m)

x R :

Position of centre of rotation (m)

x S :

Position of the solid node (m)

x W :

Position of the physical boundary (m)

α :

Fictitious collision angle(◦)

Δt :

Time step (s)

Δx :

Spatial discretisation (m)

\({\varepsilon}\) :

Porosity (−)

η :

Dynamic fluid viscosity (N s/m2)

η K :

Kolmogorov length scale (m)

ξ :

Velocity of the fluid element (m/s)

ξ σi :

Discrete velocity of the fluid element (m/s)

ρ :

Fluid density (kg/m3)

\({\tilde{\rho}}\) :

Extrapolated fluid density (kg/m3)

ρ P :

Particle density (kg/m3)

τ:

Relaxation parameter (s)

τ P :

Particle response time (s)

φ P :

Particle angular displacement (◦)

ψ :

sphericity (−)

ω P :

Particle angular velocity (1/s)

\({\boldsymbol{\omega}_P^{\ast} }\) :

Post-collision particle angular velocity (1/s)

ω σi :

Weighting factor of the discrete equilibrium distribution (−)

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Authors and Affiliations

  1. Chair of Mechanical Process Engineering, Martin-Luther-Universität Halle-Wittenberg, 06099, Halle (Saale), Germany

    Martin Ernst, Mathias Dietzel & Martin Sommerfeld

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  1. Martin Ernst
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  2. Mathias Dietzel
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Correspondence to Martin Ernst.

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Ernst, M., Dietzel, M. & Sommerfeld, M. A lattice Boltzmann method for simulating transport and agglomeration of resolved particles. Acta Mech 224, 2425–2449 (2013). https://doi.org/10.1007/s00707-013-0923-1

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