11.2 Particle-laden flows

Particle types and properties

Particle-laden flows describe systems where solid particles are carried within a moving fluid. These particles alter the overall flow behavior in ways that depend heavily on their physical characteristics. Getting a handle on particle properties is the foundation for everything else in this topic.

Three properties matter most: size, shape, and density. Each one affects how strongly the particle couples to the fluid, how much drag it experiences, and whether it tends to settle, stay suspended, or collide with other particles.

Size, shape, and density

Size controls how tightly a particle follows the fluid. Smaller particles couple more strongly to the surrounding flow, while larger particles tend to lag behind or deviate from fluid streamlines. In real systems, you rarely have one uniform particle size. The particle size distribution (PSD) captures the range and frequency of sizes present, and it can shift over time as particles agglomerate or break apart.

Shape affects drag and settling. Non-spherical particles generally experience higher drag and more complex trajectories than spheres of equivalent volume, because the drag force depends on the particle's orientation relative to the flow.

Density relative to the fluid density governs gravitational effects. A particle much denser than the fluid will tend to settle; a particle closer to the fluid density can remain suspended with relatively little effort from the flow.

Spherical vs non-spherical particles

Spherical particles are the default assumption in most models because their drag correlations and collision dynamics are well established. That simplicity is useful, but real particles are rarely perfect spheres.

Non-spherical particles (ellipsoids, fibers, irregular fragments) behave differently because their drag and lift forces depend on orientation. The aspect ratio and surface roughness both influence motion, collision probability, and agglomeration tendency.

Representing non-spherical particles in simulations requires advanced shape characterization techniques like Fourier descriptors or spherical harmonics, along with specialized collision detection algorithms. This adds significant computational cost.

Particle-fluid interactions

Particles suspended in a flow experience drag, lift, and buoyancy forces that together determine their trajectories. The strength of coupling between the particle and fluid phases depends on the particle Reynolds number, and the timescale over which a particle adjusts to changes in the flow is captured by the particle response time.

Drag force on particles

Drag is the dominant force on most suspended particles. It arises from the relative motion between the particle and the surrounding fluid and depends on:

• Particle size and shape
• Relative velocity between particle and fluid
• Fluid density and viscosity

For spherical particles, empirical correlations give the drag coefficient. Two commonly used ones are the Schiller-Naumann correlation and the Di Felice equation. For non-spherical particles, drag is typically approximated using shape factors or orientation-dependent drag models that resolve the particle's alignment with the flow.

Particle Reynolds number

The particle Reynolds number ($Re_p$) characterizes the flow regime around an individual particle:

$Re_p = \frac{\rho_f \, d_p \, |u_f - u_p|}{\mu_f}$

where $\rho_f$ is fluid density, $d_p$ is particle diameter, $u_f - u_p$ is the relative velocity between fluid and particle, and $\mu_f$ is the fluid dynamic viscosity.

• $Re_p < 1$ (Stokes regime): Viscous effects dominate. Drag is linearly proportional to relative velocity.
• $Re_p \gg 1$: Inertial effects become significant. Drag becomes nonlinear, and wake structures form behind the particle.

Particle response time

The particle response time ($\tau_p$) measures how quickly a particle adjusts to changes in the surrounding fluid velocity. For a spherical particle in the Stokes regime:

$\tau_p = \frac{\rho_p \, d_p^2}{18 \, \mu_f}$

where $\rho_p$ is the particle density.

A small $\tau_p$ means the particle tracks the fluid closely. A large $\tau_p$ means the particle has significant inertia and can deviate from fluid streamlines. The ratio of $\tau_p$ to the characteristic flow time scale gives the Stokes number, which is the central dimensionless parameter for classifying particle behavior (covered in the next section).

Particle motion in fluids

The motion of a particle in a fluid comes down to a force balance: drag, lift, buoyancy, and gravity all act simultaneously. The resulting trajectory equations are derived from Newton's second law applied to the particle.

Particle trajectory equations

The position $x_p$ and velocity $u_p$ of a particle evolve according to:

$\frac{d x_p}{dt} = u_p$

$\frac{d u_p}{dt} = \frac{f}{\tau_p}(u_f - u_p) + g$

Here, $f$ is a correction factor that accounts for non-Stokesian drag (i.e., when $Re_p$ is not small), and $g$ is gravitational acceleration.

These are ordinary differential equations solved along each particle's path. Additional terms can be included for:

• Lift forces (e.g., Saffman lift in shear flows)
• Added mass (important when $\rho_p \approx \rho_f$)
• Basset history force (accounts for the time history of relative acceleration)
• Turbulent dispersion (stochastic models for the effect of velocity fluctuations)

Stokes number and regimes

The Stokes number ($St$) is the ratio of the particle response time to the characteristic flow time scale:

$St = \frac{\tau_p}{\tau_f}$

It defines three regimes of particle behavior:

• $St \ll 1$ (equilibrium regime): Particles follow the fluid almost perfectly. Slip velocity is negligible.
• $St \gg 1$ (ballistic regime): Particles are barely influenced by the fluid and largely maintain their initial velocity.
• $St \approx 1$ (inertial regime): This is where things get interesting. Particles partially respond to the flow, leading to phenomena like preferential concentration, where particles cluster in specific regions of turbulent flows (typically in low-vorticity, high-strain-rate zones).

Gravitational settling of particles

When a particle's weight exceeds the combined buoyancy and drag forces, it settles through the fluid. In the Stokes regime ($Re_p < 1$), the terminal settling velocity of a sphere in a quiescent fluid is:

$u_t = \frac{(\rho_p - \rho_f) \, g \, d_p^2}{18 \, \mu_f}$

At higher $Re_p$, this expression must be corrected using a drag coefficient $C_D$ that accounts for inertial effects.

Gravitational settling matters in many contexts: removing particles from gas streams, sedimentation in liquid-solid suspensions, and atmospheric transport of aerosols and dust.

Turbulent particle-laden flows

Turbulence dramatically changes how particles distribute and move within a flow. Eddies can disperse particles, enhance mixing, and create non-uniform concentration patterns. At the same time, the particles themselves can modify the turbulence structure, creating a two-way coupling that complicates both experiments and simulations.

Particle dispersion in turbulence

Turbulent eddies scatter particles, generally producing a more uniform spatial distribution than laminar flow would. The key parameter is the eddy Stokes number, which compares the particle response time to the turbulent eddy time scale.

• Small eddy Stokes number: Particles are efficiently dispersed by the eddies.
• Large eddy Stokes number: Particles decouple from the eddies and tend to concentrate in low-vorticity regions (preferential concentration).

Common models for predicting particle trajectories in turbulence include the eddy interaction model (EIM) and Langevin equation models, both of which introduce stochastic velocity fluctuations along the particle path.

Particle-turbulence interactions

Particles don't just passively ride the turbulence; they feed back on it. The nature of this feedback depends on the Stokes number:

• $St \ll 1$ (small particles): Tend to attenuate turbulence by absorbing kinetic energy from the fluid through viscous dissipation.
• $St \gg 1$ (large particles): Can generate turbulence through wake effects and vortex shedding, increasing turbulent kinetic energy.
• $St \approx 1$: Particles selectively damp or enhance certain scales of motion, producing complex modulation effects.

The particle-to-fluid density ratio and the particle volume concentration also influence the strength of these interactions.

Turbulence modulation by particles

Turbulence modulation is the broader term for how particles alter the turbulent kinetic energy spectrum, dissipation rate, and characteristic length scales.

The mass loading ratio (ratio of particle to fluid mass flow rates) is a useful indicator:

• Low mass loading: Particles typically attenuate turbulence.
• High mass loading: Particles can enhance turbulence through wake generation and vortex shedding.

Numerical approaches to capture this include two-way coupled $k$-$\varepsilon$ models and equilibrium Eulerian methods, both of which add source/sink terms to the turbulence transport equations to represent the particle phase's influence.

Particle collisions and agglomeration

In dense suspensions or systems with cohesive particles, collisions and agglomeration become significant. Collisions transfer momentum, dissipate energy, and change the particle size distribution. Agglomeration (particles sticking together) can shift the effective particle size upward, altering drag, settling, and rheology.

Collision mechanisms and models

Particles can collide through several mechanisms: binary collisions in the bulk flow, wall impacts, or shear-induced contacts. The outcome of a collision depends on:

• Relative velocity and impact angle
• Particle size and material properties
• Surface forces (van der Waals, electrostatic)

Two standard modeling approaches exist:

• Hard-sphere models: Treat collisions as instantaneous events with prescribed restitution coefficients.
• Soft-sphere models: Resolve the contact over a finite time, accounting for deformation, adhesion, and frictional forces (rolling, sliding).

Agglomeration and breakup processes

Agglomeration occurs when attractive surface forces or material bonding cause colliding particles to stick together. The agglomeration rate depends on collision frequency, which in turn depends on particle concentration, size, and velocity.

Breakup happens when fluid shear forces or subsequent collisions exceed the cohesive strength of an agglomerate.

The competition between agglomeration and breakup determines how the particle size distribution evolves over time. Population balance models (PBMs) track this evolution by solving transport equations for the number density of particles in each size class, with source and sink terms for agglomeration and breakup events.

Effect on particle size distribution

Agglomeration shifts the size distribution toward larger sizes, while breakup fragments large clusters and broadens the distribution toward smaller sizes. These changes feed back into the flow:

• Larger effective particle sizes change drag and settling rates.
• A more polydisperse distribution alters the suspension's rheological properties, including viscosity and yield stress.

Accurately predicting the evolving PSD is critical in applications like fluidized beds, where particle size directly affects flow regime, heat transfer, and mixing efficiency.

Numerical methods for particle-laden flows

Simulating particle-laden flows requires methods that handle the coupling between fluid and particle phases. The right choice depends on particle concentration, Stokes number, and the level of detail you need.

Lagrangian particle tracking

Lagrangian particle tracking (LPT) solves the equation of motion for each individual particle as it moves through a pre-computed (or simultaneously computed) fluid flow field.

1. Solve the fluid flow on a fixed grid using standard CFD methods (finite volume, finite element).
2. Inject particles into the domain with initial positions and velocities.
3. At each time step, interpolate the fluid velocity at each particle's location.
4. Integrate the particle trajectory equation to update position and velocity.

LPT works well for dilute systems where particle-particle interactions are negligible. It provides detailed information about individual trajectories, residence times, and deposition patterns. The main limitation is computational cost: tracking millions of particles becomes expensive, and modeling particle-turbulence interaction requires additional sub-models.

Eulerian-Lagrangian approaches

Eulerian-Lagrangian methods extend LPT to handle moderate particle concentrations by including two-way coupling between phases. The fluid is solved on an Eulerian grid, while particles (or representative parcels of particles) are tracked in a Lagrangian frame.

Key examples include:

• DEM-CFD coupling: The discrete element method resolves individual particle contacts and collisions, coupled to a CFD solver for the fluid phase.
• MP-PIC (multiphase particle-in-cell): Groups particles into computational parcels to reduce cost while still capturing particle-fluid and particle-particle interactions.

These methods can capture complex phenomena like drafting, kissing, and tumbling (DKT) of particles in fluidized beds.

Two-fluid models and quadrature-based moments

Two-fluid models treat both the fluid and particle phases as interpenetrating continua. Instead of tracking individual particles, you solve volume-averaged Eulerian conservation equations for each phase, connected by interphase transfer terms for momentum, energy, and mass.

This approach is well suited for dense particle-laden flows where particle-particle interactions dominate. The trade-off is that you lose individual particle resolution and must rely on closure models for particle stresses and interphase exchange coefficients.

Quadrature-based moment methods (QBMM) are a specialized class of two-fluid models. Rather than solving for the full particle size distribution directly, QBMMs solve transport equations for the statistical moments of the distribution. This is computationally efficient and captures the evolution of the PSD without tracking individual particles.

Challenges common to two-fluid models include closing the interphase transfer terms, modeling particle-phase stresses (often via kinetic theory of granular flow), and resolving sharp concentration gradients numerically.

Applications of particle-laden flows

Particle-laden flows appear across a huge range of industrial and environmental settings. The physics covered above directly informs the design, optimization, and troubleshooting of real systems.

Aerosol transport and deposition

Aerosols are fine solid or liquid particles (nanometers to micrometers) suspended in a gas. Their transport is governed by the balance of fluid drag, particle inertia, and deposition mechanisms:

• Diffusion: Dominant for very small particles (sub-micron), driven by Brownian motion.
• Impaction: Particles with sufficient inertia deviate from fluid streamlines and strike surfaces.
• Interception: Particles following streamlines pass close enough to a surface to make contact.

Numerical predictions use either the advection-diffusion equation (Eulerian) or Lagrangian particle tracking. Applications include atmospheric pollutant dispersion, inhaler design for respiratory drug delivery, and contamination control in semiconductor cleanrooms.

Fluidized beds and pneumatic conveying

In a fluidized bed, an upward fluid flow suspends a bed of solid particles, creating fluid-like behavior. This provides excellent heat and mass transfer, uniform temperatures, and thorough mixing. Flow regimes range from bubbling to slugging to turbulent fluidization, depending on gas velocity and particle properties.

Pneumatic conveying transports solid particles through pipes using a carrier gas. It's widely used in bulk material handling and powder processing.

Both systems require understanding of particle-fluid interactions, PSD effects, and flow regime transitions. Simulations typically use two-fluid models or DEM-CFD coupling to predict hydrodynamics, mixing, and heat transfer.

Spray drying and atomization processes

Spray drying converts liquid solutions or suspensions into dry powder by atomizing the liquid into fine droplets and exposing them to hot gas. Atomization is the breakup of a liquid stream into droplets, which is also central to combustion applications.

Performance depends on nozzle design, liquid properties (viscosity, surface tension), gas flow conditions, and the drying kinetics that govern final particle size and moisture content.

Atomization models like the Kelvin-Helmholtz/Rayleigh-Taylor (KHRT) breakup model and the Eulerian-Lagrangian spray atomization (ELSA) approach simulate liquid jet breakup and droplet formation. Applications span food powder production, pharmaceuticals, ceramics, and fuel injection in combustion systems.