11.2 Particle-laden flows

Particle-laden flows involve solid particles moving within fluids, affecting overall flow behavior. Understanding particle properties like size, shape, and density is key to modeling these flows accurately. This knowledge is crucial for applications in fluid dynamics and engineering.

Particles interact with fluids through forces like drag and lift, influencing their motion and trajectory. The particle Reynolds number and response time determine how particles behave in different flow conditions. These concepts are essential for predicting particle behavior in various fluid systems.

Particle types and properties

• Particle-laden flows involve the transport and interaction of solid particles within a fluid medium, which can significantly impact the overall flow behavior and characteristics
• Understanding the properties of particles, such as size, shape, and density, is crucial for accurately modeling and predicting their behavior in various fluid dynamics applications
• Particle properties can influence the drag force experienced by particles, their response to fluid motion, and their tendency to collide and agglomerate

Size, shape, and density

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• Particle size is a key parameter that affects particle-fluid interactions, with smaller particles generally exhibiting stronger coupling with the fluid flow
• Particle size distribution (PSD) characterizes the range and frequency of particle sizes present in a system, which can evolve due to processes like agglomeration and breakup
• Particle shape influences the drag force and settling behavior, with non-spherical particles often experiencing higher drag and more complex motion compared to spherical particles
• Particle density relative to the fluid density determines the importance of gravitational effects and the tendency for particles to settle or become suspended in the flow

Spherical vs non-spherical particles

• Spherical particles are often assumed for simplicity in modeling and theoretical analysis, as they have well-defined drag correlations and collision dynamics
• Non-spherical particles, such as ellipsoids, fibers, or irregular shapes, exhibit more complex behavior due to their orientation-dependent drag and lift forces
• The aspect ratio and surface roughness of non-spherical particles can significantly impact their motion, collision probability, and agglomeration tendencies
• Accurately representing non-spherical particles in numerical simulations often requires advanced shape characterization techniques (Fourier descriptors, spherical harmonics) and specialized collision detection algorithms

Particle-fluid interactions

• Particles in a fluid flow experience various forces, including drag, lift, and buoyancy, which govern their motion and trajectory
• The coupling between particles and the fluid is characterized by the particle Reynolds number, which determines the relative importance of inertial and viscous effects
• Particle-fluid interactions can lead to complex phenomena, such as preferential concentration in turbulent flows and modification of the fluid flow field

Drag force on particles

• Drag force is the primary force acting on particles due to the relative motion between the particle and the surrounding fluid
• The drag force depends on the particle size, shape, and relative velocity, as well as the fluid properties (density, viscosity)
• Empirical correlations, such as the Schiller-Naumann or Di Felice equations, are commonly used to estimate the drag coefficient for spherical particles
• For non-spherical particles, the drag force is often approximated using shape factors or by resolving the particle orientation and applying orientation-dependent drag models

Particle Reynolds number

• The particle Reynolds number ($Re_p$) is a dimensionless quantity that characterizes the flow regime around a particle
• $Re_p$ is defined as $Re_p = \frac{\rho_f d_p |u_f - u_p|}{\mu_f}$, where $\rho_f$ is the fluid density, $d_p$ is the particle diameter, $u_f$ and $u_p$ are the fluid and particle velocities, and $\mu_f$ is the fluid dynamic viscosity
• At low $Re_p$ ($Re_p < 1$), the flow around the particle is dominated by viscous effects, and the drag force is linearly proportional to the relative velocity (Stokes regime)
• At higher $Re_p$, inertial effects become significant, leading to nonlinear drag and the formation of wake structures behind the particles

Particle response time

• The particle response time ($\tau_p$) is a measure of how quickly a particle responds to changes in the surrounding fluid velocity
• For spherical particles, $\tau_p$ is given by $\tau_p = \frac{\rho_p d_p^2}{18 \mu_f}$, where $\rho_p$ is the particle density
• Particles with smaller response times tend to follow the fluid flow more closely, while particles with larger response times exhibit more inertial behavior and can deviate from the fluid streamlines
• The ratio of the particle response time to the characteristic flow time scale (Stokes number) determines the degree of coupling between the particles and the fluid flow

Particle motion in fluids

• The motion of particles in fluids is governed by the balance of forces acting on them, including drag, lift, buoyancy, and gravity
• Particle trajectory equations, derived from Newton's second law, describe the temporal evolution of particle position and velocity
• The Stokes number, which compares the particle response time to the characteristic flow time scale, determines the regime of particle motion (equilibrium, inertial, or ballistic)

Particle trajectory equations

• Particle trajectory equations are ordinary differential equations that describe the position ($x_p$) and velocity ($u_p$) of a particle as a function of time
• The basic form of the trajectory equations for spherical particles is:
  • $\frac{d x_p}{dt} = u_p$
  • $\frac{d u_p}{dt} = \frac{f}{\tau_p} (u_f - u_p) + g$
• Here, $f$ is a correction factor accounting for non-Stokesian drag, and $g$ is the gravitational acceleration
• Additional terms can be included to account for other forces (lift, added mass, Basset history term) or to model the effect of turbulent fluctuations on particle motion

Stokes number and regimes

• The Stokes number ($St$) is a dimensionless parameter that characterizes the behavior of particles in a fluid flow
• $St$ is defined as the ratio of the particle response time ($\tau_p$) to the characteristic flow time scale ($\tau_f$): $St = \frac{\tau_p}{\tau_f}$
• For $St << 1$ (equilibrium regime), particles closely follow the fluid flow and have negligible slip velocity
• For $St >> 1$ (ballistic regime), particles are largely unaffected by the fluid flow and maintain their initial velocity
• In the intermediate range ($St \approx 1$, inertial regime), particles exhibit complex behavior, such as preferential concentration and clustering in turbulent flows

Gravitational settling of particles

• Gravitational settling occurs when the weight of a particle exceeds the buoyancy and drag forces, causing it to fall through the fluid
• The terminal settling velocity ($u_t$) of a spherical particle in a quiescent fluid is given by $u_t = \frac{(\rho_p - \rho_f) g d_p^2}{18 \mu_f}$ for Stokes flow ($Re_p < 1$)
• For higher particle Reynolds numbers, the settling velocity is modified by a drag correction factor ($C_D$) that accounts for inertial effects
• Gravitational settling is important in applications such as particle removal from gases, sedimentation in liquid-solid suspensions, and the transport of aerosols in the atmosphere

Turbulent particle-laden flows

• Turbulent particle-laden flows are characterized by the complex interaction between particles and the fluctuating fluid velocity field
• Particles can be dispersed by turbulent eddies, leading to enhanced mixing and transport compared to laminar flows
• The presence of particles can also modify the turbulence structure, either suppressing or enhancing turbulent fluctuations depending on the particle size, concentration, and flow conditions

Particle dispersion in turbulence

• Turbulent eddies can disperse particles, leading to a more homogeneous distribution compared to laminar flows
• The degree of particle dispersion depends on the ratio of the particle response time to the turbulent eddy time scale (eddy Stokes number)
• For small eddy Stokes numbers, particles are efficiently dispersed by the turbulent flow, while for large eddy Stokes numbers, particles may concentrate in low-vorticity regions
• Particle dispersion models, such as the eddy interaction model (EIM) or the Langevin equation model, are used to predict particle trajectories in turbulent flows

Particle-turbulence interactions

• Particles can modify the turbulence structure through various mechanisms, such as turbulence attenuation, generation, or modulation
• Small particles ($St << 1$) tend to attenuate turbulence by absorbing energy from the fluid through viscous dissipation
• Large particles ($St >> 1$) can generate turbulence in their wake or through vortex shedding, leading to increased turbulent kinetic energy
• In the intermediate range ($St \approx 1$), particles can modulate turbulence by selectively damping or enhancing certain scales of motion
• The extent of particle-turbulence interaction depends on factors such as particle size, concentration, and the ratio of particle to fluid density

Turbulence modulation by particles

• Turbulence modulation refers to the modification of turbulent flow characteristics by the presence of particles
• Particles can alter the turbulent kinetic energy spectrum, energy dissipation rate, and length scales of turbulence
• The modulation effect depends on the particle Stokes number and the mass loading ratio (ratio of particle to fluid mass flow rates)
• For low mass loading ratios, particles typically attenuate turbulence, while for high mass loading ratios, particles can enhance turbulence through wake generation and vortex shedding
• Turbulence modulation models, such as the two-way coupled k-ε model or the equilibrium Eulerian approach, are used to account for the effects of particles on turbulence in numerical simulations

Particle collisions and agglomeration

• Particle collisions and agglomeration are important phenomena in particle-laden flows, particularly in dense suspensions or when particle cohesion is significant
• Collisions between particles can lead to momentum transfer, energy dissipation, and changes in particle size distribution due to agglomeration or breakup
• Agglomeration, the process by which particles stick together to form larger clusters, can significantly impact the flow behavior and rheological properties of the suspension

Collision mechanisms and models

• Particle collisions can occur due to various mechanisms, such as binary collisions, wall collisions, or collisions in shear flows
• The outcome of a collision depends on factors like the particle velocity, size, material properties, and surface forces (van der Waals, electrostatic)
• Collision models, such as the hard-sphere or soft-sphere models, are used to predict the post-collision velocities and contact forces between particles
• More advanced collision models account for factors such as particle deformation, adhesion, and rolling or sliding friction

Agglomeration and breakup processes

• Agglomeration occurs when particles collide and stick together due to attractive surface forces or material bonding
• The rate of agglomeration depends on the collision frequency, which is influenced by factors like particle concentration, size, and velocity
• Breakup of agglomerates can occur due to fluid shear forces or collisions with other particles or walls
• The balance between agglomeration and breakup processes determines the evolution of the particle size distribution in the system
• Population balance models (PBMs) are used to predict the temporal evolution of the particle size distribution considering agglomeration and breakup events

Effect on particle size distribution

• Particle collisions and agglomeration can significantly alter the particle size distribution in a particle-laden flow
• Agglomeration leads to the formation of larger particles or clusters, shifting the size distribution towards larger sizes
• Breakup events can fragment large agglomerates into smaller particles, resulting in a more polydisperse size distribution
• The evolution of the particle size distribution can impact the rheological properties of the suspension, such as viscosity and yield stress
• Accurately predicting the particle size distribution is crucial for applications like fluidized beds, where the size distribution affects the flow behavior and heat transfer characteristics

Numerical methods for particle-laden flows

• Numerical simulations of particle-laden flows require specialized methods to handle the coupling between the fluid and particle phases
• The choice of numerical approach depends on factors such as the particle concentration, Stokes number, and the desired level of resolution
• Common numerical methods for particle-laden flows include Lagrangian particle tracking, Eulerian-Lagrangian approaches, and two-fluid models

Lagrangian particle tracking

• Lagrangian particle tracking (LPT) is a method where individual particle trajectories are computed by solving the particle equation of motion
• The fluid flow field is first solved using a conventional CFD approach (finite volume, finite element), and then particles are tracked through the flow domain
• LPT is suitable for dilute systems with low particle concentrations, where particle-particle interactions are negligible
• The method can provide detailed information about particle trajectories, residence times, and deposition patterns
• Challenges in LPT include the computational cost for large numbers of particles and the modeling of particle-turbulence interactions

Eulerian-Lagrangian approaches

• Eulerian-Lagrangian approaches combine a Eulerian description of the fluid phase with a Lagrangian tracking of particles
• The fluid flow is solved using Eulerian conservation equations, while particles are tracked individually or as parcels using Lagrangian equations of motion
• Eulerian-Lagrangian methods can handle higher particle concentrations compared to pure LPT, as they account for particle-fluid interactions and two-way coupling
• Examples of Eulerian-Lagrangian approaches include the discrete element method (DEM) coupled with CFD and the multiphase particle-in-cell (MP-PIC) method
• These methods can capture complex particle-fluid interactions, such as drafting, kissing, and tumbling (DKT) in fluidized beds

Two-fluid models and quadrature-based moments

• Two-fluid models treat both the fluid and particle phases as interpenetrating continua, solving Eulerian conservation equations for each phase
• The particle phase is described by volume-averaged equations, accounting for particle-fluid interactions through interphase transfer terms
• Two-fluid models are suitable for dense particle-laden flows, where particle-particle interactions and collisions are significant
• Quadrature-based moment methods (QBMM) are a class of two-fluid models that solve transport equations for the moments of the particle size distribution
• QBMMs provide a computationally efficient way to capture the evolution of the particle size distribution without the need to track individual particles
• Challenges in two-fluid models include the closure of interphase transfer terms, the modeling of particle stresses, and the numerical resolution of particle concentration gradients

Applications of particle-laden flows

• Particle-laden flows are encountered in a wide range of industrial and environmental applications, spanning various length and time scales
• Understanding and predicting the behavior of particle-laden flows is crucial for process optimization, product quality control, and environmental impact assessment
• Some key applications of particle-laden flows include aerosol transport and deposition, fluidized beds, pneumatic conveying, spray drying, and atomization processes

Aerosol transport and deposition

• Aerosol transport and deposition are important in fields such as air pollution, respiratory drug delivery, and clean room technology
• Aerosols are fine solid or liquid particles suspended in a gas, with sizes ranging from nanometers to micrometers
• The transport of aerosols is governed by the interplay of fluid flow, particle inertia, and various deposition mechanisms (diffusion, impaction, interception)
• Numerical models, such as the advection-diffusion equation or Lagrangian particle tracking, are used to predict aerosol concentration fields and deposition patterns
• Applications include predicting the dispersion of pollutants in the atmosphere, optimizing inhaler designs for targeted drug delivery, and controlling particle contamination in semiconductor manufacturing

Fluidized beds and pneumatic conveying

• Fluidized beds are widely used in chemical processing, energy generation, and granular material handling
• In a fluidized bed, a vertical upward flow of fluid (gas or liquid) suspends solid particles, creating a fluid-like behavior of the particle bed
• Fluidized beds offer advantages such as enhanced heat and mass transfer, uniform temperature distribution, and good mixing of the particulate phase
• Pneumatic conveying is a method of transporting solid particles through pipes using a carrier gas, often used in bulk material handling and powder processing
• The design and operation of fluidized beds and pneumatic conveying systems require an understanding of particle-fluid interactions, particle size distribution, and flow regimes (bubbling, slugging, turbulent)
• Numerical simulations, such as two-fluid models or DEM-CFD coupling, are used to predict the hydrodynamics, mixing, and heat transfer in fluidized beds and pneumatic conveying systems

Spray drying and atomization processes

• Spray drying is a widely used method for converting liquid solutions or suspensions into dry powder form
• In spray drying, the liquid feed is atomized into fine droplets, which are then dried by hot gas in a drying chamber
• Atomization is the process of breaking up a liquid into small droplets, which is crucial for efficient spray drying and combustion applications
• The performance of spray drying and atomization processes depends on factors such as the nozzle design, liquid properties, gas flow conditions, and particle formation mechanisms
• Numerical modeling of spray drying involves coupling the gas flow, droplet trajectories, and drying kinetics to predict the final particle size distribution and moisture content
• Atomization models, such as the Kelvin-Helmholtz and Rayleigh-Taylor (KHRT) breakup model or the Eulerian-Lagrangian spray atomization (ELSA) approach, are used to simulate the liquid jet breakup and droplet formation in atomization processes
• Applications of spray drying and atomization include the production of food powders, pharmaceuticals, and ceramic materials, as well as the design of combust