is a crucial concept in multiphase flow modeling. It opposes the motion of objects through fluids, affecting the behavior of particles, bubbles, and droplets in various industrial applications. Understanding drag force is essential for accurately predicting the dynamics of dispersed phases in multiphase systems.
This topic covers drag force fundamentals, single particle drag, models for particle swarms, bubbles, and droplets, and effects of turbulence. It also discusses numerical implementation, advanced topics, and validation of drag models. These concepts are vital for simulating complex multiphase flows in engineering applications.
Drag force fundamentals
Drag force is a resistive force that opposes the motion of an object through a fluid
Fundamentals of drag force are essential in understanding and modeling multiphase flows, where particles, bubbles, or droplets interact with a continuous fluid phase
Accurate prediction of drag force is crucial for simulating the behavior and transport of dispersed phases in various industrial applications (fluidized beds, bubble columns, sprays)
Definition of drag force
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Numerical simulation of multiphase flow in a Vanyukov furnace View original
Drag force governs the motion and distribution of dispersed phases (particles, bubbles, droplets) in multiphase flows
Influences key phenomena such as settling, entrainment, mixing, and separation
Accurate modeling of drag force is essential for predicting the hydrodynamics, heat and mass transfer, and chemical reactions in multiphase systems
Drag force on a single particle
Understanding drag force on a single particle forms the basis for modeling drag in more complex multiphase systems
Single particle drag models are used as building blocks for developing drag correlations for particle swarms, bubbles, and droplets
Stokes' law for creeping flow
Stokes' law describes the drag force on a spherical particle in creeping flow (low , Re<<1)
Drag force is given by FD=3πμdpv, where μ is fluid viscosity, dp is particle diameter, and v is relative velocity
Assumes no fluid inertia and no slip condition at the particle surface
Drag coefficient vs Reynolds number
Drag coefficient CD is a dimensionless quantity that relates the drag force to the fluid dynamic pressure and the
CD depends on the particle Reynolds number Rep=μρdpv, which characterizes the ratio of inertial to viscous forces
For creeping flow (Rep<<1), CD=Rep24 (Stokes' law)
As Rep increases, CD deviates from Stokes' law due to fluid inertia and boundary layer separation
Effect of particle shape
Non-spherical particles experience higher drag compared to spherical particles of the same volume
Shape effects are often accounted for by introducing a shape factor or an equivalent diameter in the drag correlation
Common shape factors include sphericity (ratio of surface area of equivalent sphere to actual surface area) and aspect ratio (ratio of longest to shortest dimension)
Drag models for particle swarms
In many multiphase flows, particles exist as swarms or clusters rather than isolated entities
Particle-particle interactions and the presence of neighboring particles modify the drag force experienced by individual particles
Particle-particle interactions
Particles in a swarm interact through hydrodynamic forces (drag, lift) and collisions
Hydrodynamic interactions lead to hindered settling, where the settling velocity of a particle is reduced compared to an isolated particle
Collisions result in momentum exchange and can promote or hinder particle motion depending on the flow conditions
Empirical drag correlations
Empirical drag correlations are developed based on experimental data or numerical simulations to account for the effects of particle swarms
Examples include for hindered settling and for dilute particle suspensions
These correlations modify the single particle drag coefficient by introducing a function of the particle volume fraction αp
Ergun equation for packed beds
is a widely used drag model for dense particle beds, where particles are in contact with each other
Combines the viscous (Kozeny-Carman) and inertial (Burke-Plummer) contributions to the pressure drop
Drag force is expressed as a function of the fluid velocity, bed voidage, and particle size and shape
Applicable to fixed and fluidized beds, as well as porous media flows
Drag models for bubbles and droplets
Bubbles and droplets are fluid particles that deform and interact differently compared to solid particles
Drag models for bubbles and droplets account for the effects of deformation, internal circulation, and surface tension
Schiller-Naumann correlation
is a simple drag model for spherical bubbles and droplets in the Stokes and moderate Reynolds number regimes
Drag coefficient is given by CD=Re24(1+0.15Re0.687), where Re is the bubble/droplet Reynolds number
Valid for Re<1000 and assumes no deformation or internal circulation
Ishii-Zuber drag model
accounts for the deformation of bubbles and droplets in viscous and turbulent flows
Introduces a modified Reynolds number that incorporates the Eötvös number (ratio of buoyancy to surface tension forces) and the Morton number (ratio of viscous to surface tension forces)
Provides separate expressions for the drag coefficient in the viscous, distorted, and churn-turbulent regimes
Effect of bubble/droplet deformation
Deformation of bubbles and droplets alters their shape and increases their projected area, leading to higher drag forces
Deformation is influenced by the balance between fluid stresses and surface tension forces
Drag models incorporating deformation effects use shape factors or equivalent diameters based on the deformed geometry
Examples include the Grace model for bubbles and the Henschke model for droplets
Drag in turbulent flows
Turbulence adds complexity to the modeling of drag force in multiphase flows
Turbulent eddies enhance the dispersion and mixing of particles, bubbles, and droplets, and modify the drag force experienced by them
Turbulent dispersion force
Turbulent dispersion force represents the effect of turbulent eddies on the motion of dispersed phases
Acts to redistribute the dispersed phase from high concentration regions to low concentration regions
Modeled using a gradient diffusion hypothesis, where the dispersion force is proportional to the gradient of the dispersed phase volume fraction
Eddy-particle interaction time
Eddy-particle interaction time τep characterizes the time scale of interaction between turbulent eddies and dispersed particles
Determines the relative importance of turbulent dispersion and particle inertia
Defined as the minimum of the eddy lifetime and the particle response time
Used in stochastic drag models and turbulent dispersion force formulations
Stochastic drag models
Stochastic drag models account for the fluctuating nature of turbulent flows by introducing random variations in the drag force
Examples include the Langevin equation model and the discrete random walk model
Stochastic models simulate the instantaneous velocity and position of particles, bubbles, or droplets by solving a set of stochastic differential equations
Provide a more realistic representation of turbulence-particle interactions compared to deterministic models
Numerical implementation of drag models
Drag models are implemented in codes to simulate multiphase flows
Numerical implementation involves discretizing the governing equations and coupling the drag force with the momentum equations
Drag force term in momentum equations
Drag force appears as a source term in the momentum equations for the dispersed and continuous phases
Momentum exchange between phases is governed by the drag force, ensuring conservation of momentum
Drag force term is typically formulated as a function of the relative velocity and the drag coefficient
Explicit vs implicit treatment
Drag force can be treated explicitly or implicitly in the numerical scheme
Explicit treatment calculates the drag force using the values from the previous time step, while implicit treatment uses the values from the current time step
Implicit treatment is more stable and allows for larger time steps, but requires solving a coupled system of equations
Stability and convergence issues
Numerical stability and convergence are important considerations in the implementation of drag models
Explicit treatment of drag force can lead to instability if the time step is too large relative to the particle relaxation time
Implicit treatment improves stability but may suffer from convergence issues if the coupling between phases is strong
Techniques such as under-relaxation, time step adaptation, and multigrid methods can be used to enhance stability and convergence
Advanced topics in drag modeling
Drag modeling in multiphase flows is an active area of research, with ongoing developments to capture complex physics and improve accuracy
Near-wall drag modifications
Presence of walls modifies the drag force on particles, bubbles, and droplets due to the formation of boundary layers and wall-induced turbulence
Near-wall drag models account for the enhanced drag and lift forces experienced by dispersed phases in the vicinity of walls
Examples include the for bubbles and the for particles
Drag in polydisperse mixtures
Polydisperse mixtures contain dispersed phases with a range of sizes, shapes, or densities
Drag models for polydisperse mixtures need to account for the interactions between different size classes or types of dispersed phases
Approaches include the multi-fluid model, where each size class is treated as a separate phase, and the population balance model, which tracks the size distribution evolution
Drag in non-Newtonian fluids
Non-Newtonian fluids exhibit complex rheological behavior, such as shear-thinning or shear-thickening, which affects the drag force on dispersed phases
Drag models for non-Newtonian fluids incorporate the fluid rheology through the use of apparent or effective viscosities
Examples include the Ishii-Zuber model for bubbles in power-law fluids and the Renaud model for particles in Bingham plastics
Validation and limitations of drag models
Validation of drag models is essential to assess their accuracy and applicability to different multiphase flow scenarios
Limitations of drag models should be recognized to ensure their appropriate use and interpretation
Comparison with experimental data
Drag models are validated by comparing their predictions with experimental measurements of key quantities (pressure drop, phase velocities, volume fractions)
Experimental techniques used for validation include , laser Doppler anemometry (LDA), and electrical capacitance tomography (ECT)
Good agreement between model predictions and experimental data builds confidence in the drag model's accuracy
Range of applicability
Drag models are developed based on certain assumptions and have a limited range of applicability
Factors that limit the applicability of drag models include the flow regime (laminar, transitional, turbulent), the dispersed phase properties (size, shape, concentration), and the fluid properties (viscosity, density, surface tension)
Drag models should be selected based on their suitability for the specific multiphase flow conditions under consideration
Best practices for drag model selection
Choose drag models that are appropriate for the flow regime, dispersed phase characteristics, and fluid properties
Consider the level of complexity required (single particle, swarm, deformable bubbles/droplets) and the computational cost
Validate the selected drag model against experimental data or benchmark simulations for similar flow conditions
Perform sensitivity analysis to assess the impact of drag model parameters and uncertainties on the simulation results
Document the drag model selection process and justify the choices made based on the available evidence and best practices in the field
Key Terms to Review (24)
Antal-Lahey-Flaherty Model: The Antal-Lahey-Flaherty Model is a mathematical framework used to describe the drag force acting on solid particles in a multiphase flow, specifically in gas-solid systems. This model integrates particle shape, size, and flow characteristics to predict how particles will move through a fluid medium, providing insights into the behavior of such systems under various conditions.
Computational Fluid Dynamics (CFD): Computational Fluid Dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and algorithms to solve and analyze problems involving fluid flows. This technique is essential for simulating the behavior of multiphase flows, allowing engineers to predict flow patterns, heat transfer, and chemical reactions in various applications, from reactors to pipelines.
Drag Coefficient Equation: The drag coefficient equation is a mathematical formula that quantifies the drag force acting on an object moving through a fluid, allowing for the prediction of resistance faced by the object. It incorporates various factors such as shape, size, and flow conditions, making it a critical element in understanding drag force and developing drag models in fluid dynamics. By relating drag force to fluid density, velocity, and the object's reference area, it helps engineers and scientists analyze performance in diverse applications, from aerodynamics to hydraulics.
Drag Force: Drag force is the resistance force experienced by an object moving through a fluid, resulting from the interaction between the object's surface and the fluid molecules. This force plays a crucial role in multiphase flows, influencing how particles or droplets behave as they move through gases or liquids, and it is essential in understanding various phenomena such as momentum transfer, sediment transport, and the dynamics of fluidized bed reactors.
Drag force equation: The drag force equation quantifies the resistance experienced by an object moving through a fluid, which is essential for understanding how multiphase flows behave. This equation relates the drag force to factors such as the object's velocity, fluid density, drag coefficient, and cross-sectional area. Understanding this relationship is crucial for modeling the dynamics of particles and droplets within multiphase systems, as it impacts their motion and interactions.
Drag partitioning theory: Drag partitioning theory explains how the total drag force acting on a particle in a multiphase flow can be divided into different components that represent the contributions of various phases. This theory helps in understanding the complex interactions between solid particles and the surrounding fluid phases, leading to better predictions of motion and behavior in a flow system.
Ergun Equation: The Ergun equation is a fundamental equation used to calculate the pressure drop across a packed bed of particles when fluid flows through it. It combines both viscous and inertial effects of the fluid, making it essential for understanding flow behavior in various multiphase systems. This equation plays a crucial role in predicting drag force and characterizing flow regimes, especially in applications involving trickle bed reactors and fluidized bed reactors.
Finite Volume Method: The finite volume method is a numerical technique used for solving partial differential equations, particularly in fluid dynamics, by dividing the domain into small control volumes. This approach helps in conserving mass, momentum, and energy by integrating these quantities over each control volume and applying the principles of flux across the boundaries. It connects well with various models and transfer processes involved in multiphase flows, as it efficiently handles complex geometries and varying flow conditions.
Force balance technique: The force balance technique is a method used to analyze the forces acting on particles within a multiphase flow system to determine their behavior and interactions. This technique plays a crucial role in understanding how drag force affects particle motion, allowing for the development of predictive models for particle transport and sedimentation in various fluid dynamics scenarios.
Interfacial Tension: Interfacial tension is the force that exists at the interface between two immiscible fluids, which acts to minimize the surface area and create a stable boundary between the fluids. This phenomenon plays a crucial role in various multiphase flow dynamics, affecting how different phases interact, disperse, and behave under various conditions.
Ishii-Zuber Drag Model: The Ishii-Zuber Drag Model is a mathematical framework used to quantify the drag force acting on dispersed phases in a multiphase flow, particularly focusing on gas-liquid systems. This model provides a way to predict how different phases interact with each other and the impact of these interactions on the overall flow behavior. Understanding this model is crucial for accurately modeling multiphase systems, especially in applications like chemical reactors and heat exchangers.
Lift force: Lift force refers to the net force acting on a particle or bubble in a multiphase flow that acts perpendicular to the direction of the flow due to pressure differences. It plays a crucial role in interphase momentum transfer, helping to determine how particles or droplets behave within a fluid medium. Lift force is essential in understanding drag forces and models, as it influences the overall motion and stability of particles, while also being tied to virtual mass forces, which account for the inertia of displaced fluid surrounding moving particles.
McLaughlin Model: The McLaughlin Model is a mathematical representation used to describe the drag force acting on particles in a multiphase flow system, particularly focusing on the influence of particle shape and size on drag. This model is essential in understanding how particles interact with the fluid around them, which can significantly affect the overall flow dynamics and performance of processes such as sediment transport and chemical reactions.
Newtonian Drag Model: The Newtonian drag model is a mathematical representation that describes the drag force acting on a particle moving through a fluid, based on Newton's second law of motion. This model assumes that the drag force is proportional to the velocity of the particle, which is valid for certain flow regimes, particularly in laminar flow where viscous effects dominate. Understanding this model is crucial for predicting how particles behave in multiphase flow scenarios, allowing engineers and scientists to design systems more effectively.
Particle image velocimetry (PIV): Particle image velocimetry (PIV) is an optical method used to measure the velocity of fluid flow by capturing images of tracer particles suspended in the fluid. This technique allows researchers to visualize flow patterns and obtain quantitative data on velocity fields, making it essential for studying various multiphase flow phenomena and enhancing our understanding of complex interactions between phases.
Particle size: Particle size refers to the diameter or dimensions of individual particles within a multiphase system. This term is crucial as it influences the behavior of particles in fluid flow, particularly how they interact with the surrounding fluid and affect drag forces, which are pivotal in modeling multiphase flow systems.
Pressure drag: Pressure drag is a type of aerodynamic or hydrodynamic drag force experienced by an object moving through a fluid, resulting from the pressure differential between the front and rear surfaces of the object. This force occurs due to the shape and orientation of the object, which influences how fluid flows around it, leading to changes in pressure distribution and increased resistance against motion.
Reynolds Number: Reynolds number is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. It is calculated using the ratio of inertial forces to viscous forces, providing insight into whether the flow will be laminar or turbulent. Understanding Reynolds number is crucial for analyzing fluid behavior in various systems, such as flow pattern maps, drag forces, stirred tank reactors, condensers, distillation columns, and liquid-liquid flow regimes.
Richardson-Zaki Correlation: The Richardson-Zaki correlation is an empirical relationship used to estimate the drag force acting on solid particles suspended in a fluid, particularly in the context of fluidized systems. This correlation is significant for predicting how particles behave when they are suspended in a fluid, providing insights into the transition from packed beds to fully fluidized states and influencing the design of various multiphase flow systems.
Schiller-Naumann Correlation: The Schiller-Naumann correlation is a mathematical relationship used to estimate the drag force acting on particles in a multiphase flow. This correlation specifically addresses the drag coefficient of particles as a function of their Reynolds number, incorporating the effects of particle shape and fluid properties. It is crucial for accurately predicting the motion and behavior of particles suspended in a fluid, making it an essential concept in drag force modeling.
Slip Velocity: Slip velocity is the relative velocity between phases in a multiphase flow, typically describing the motion of dispersed particles or droplets relative to the surrounding continuous phase. Understanding slip velocity is crucial for predicting how different phases interact and move within a flow, influencing aspects like momentum transfer, drag force, and overall flow behavior.
Stokes Drag Model: The Stokes Drag Model describes the drag force experienced by small spherical particles moving through a viscous fluid at low Reynolds numbers. This model is essential in understanding the motion of particles in multiphase flows, particularly in scenarios where inertial effects are negligible and viscous forces dominate.
Turbulence modeling: Turbulence modeling refers to the mathematical and computational techniques used to simulate and predict the behavior of turbulent flows, which are characterized by chaotic changes in pressure and flow velocity. It is essential for understanding complex multiphase flows, as it helps capture the interactions between different phases and the impact of turbulence on transport phenomena, such as momentum and mass transfer. Effective turbulence models are vital for accurately representing the dynamics of fluids in various applications.
Wen-Yu Correlation: The Wen-Yu correlation is a specific empirical relationship used to estimate the drag force experienced by solid particles moving through a fluid. This correlation is significant for modeling multiphase flows, as it provides a way to relate the drag coefficient to particle Reynolds number and other factors affecting fluid-particle interactions, facilitating more accurate predictions in drag force calculations.