💧Multiphase Flow Modeling Unit 5 – Momentum, Heat & Mass Transfer in Multiphase Flow

Key Concepts and Definitions

• Multiphase flow involves the simultaneous flow of two or more phases (gas, liquid, or solid) in a system
• Phase refers to a distinct state of matter with uniform physical and chemical properties
• Interfacial area is the surface area between different phases where transfer processes occur
• Volume fraction represents the proportion of each phase in a given volume of the multiphase system
• Slip velocity describes the relative velocity between different phases in a multiphase flow
• Dispersed phase consists of discrete elements (bubbles, droplets, or particles) distributed within a continuous phase
  • Examples include gas bubbles in a liquid (bubbly flow) or solid particles in a fluid (slurry flow)
• Continuous phase is the phase that forms a connected and continuous medium surrounding the dispersed phase elements

Fundamental Equations and Models

• Conservation equations form the basis for describing multiphase flow behavior
  • Mass conservation equation ensures that mass is neither created nor destroyed in the system
  • Momentum conservation equation describes the balance of forces acting on each phase
  • Energy conservation equation accounts for heat transfer and energy exchange between phases
• Interphase transfer terms in the conservation equations represent the exchange of mass, momentum, and energy between phases
• Constitutive relations provide closure to the conservation equations by relating unknown variables to known quantities
  • Examples include drag force models, heat transfer coefficients, and mass transfer correlations
• Averaging techniques (volume averaging or ensemble averaging) are used to derive macroscopic equations for multiphase flow
• Two-fluid model treats each phase as a separate fluid with its own set of conservation equations coupled through interphase transfer terms
• Mixture model considers the multiphase system as a single fluid with averaged properties and additional terms to account for interphase interactions

Momentum Transfer in Multiphase Systems

• Drag force is the resistance experienced by a dispersed phase element moving through the continuous phase
  • Depends on the relative velocity, fluid properties, and particle shape and size
• Virtual mass force accounts for the acceleration of the continuous phase surrounding the dispersed phase elements
• Lift force acts perpendicular to the relative velocity and is caused by the asymmetric pressure distribution around the dispersed phase elements
• Turbulent dispersion force represents the effect of turbulent fluctuations on the dispersed phase distribution
• Basset force is a history-dependent force that accounts for the temporal delay in boundary layer development around accelerating particles
• Particle-particle interactions (collisions and coalescence) can significantly influence the momentum transfer in dense multiphase flows
• Effective viscosity models are used to describe the rheological behavior of multiphase mixtures
  • Examples include the Einstein model for dilute suspensions and the Krieger-Dougherty model for concentrated suspensions

Heat Transfer Mechanisms

• Conduction is the transfer of heat through a medium due to temperature gradients without bulk motion of the medium
  • Governed by Fourier's law, which relates heat flux to temperature gradient and thermal conductivity
• Convection involves the transfer of heat between a surface and a moving fluid
  • Forced convection occurs when the fluid motion is induced by external means (pumps or fans)
  • Natural convection is driven by buoyancy forces arising from density differences caused by temperature variations
• Radiation is the transfer of heat through electromagnetic waves without the need for an intervening medium
  • Becomes significant at high temperatures or in systems with large temperature differences
• Interfacial heat transfer occurs between different phases due to temperature differences at the interface
  • Characterized by heat transfer coefficients that depend on the flow regime, fluid properties, and interface geometry
• Effective thermal conductivity models are used to describe the overall heat transfer in multiphase systems
  • Account for the contributions of different phases and their distribution

Mass Transfer Processes

• Diffusion is the transport of species due to concentration gradients without bulk motion of the fluid
  • Described by Fick's law, which relates mass flux to concentration gradient and diffusion coefficient
• Convective mass transfer involves the transport of species by the bulk motion of the fluid
  • Analogous to convective heat transfer and characterized by mass transfer coefficients
• Interfacial mass transfer occurs between different phases due to concentration differences at the interface
  • Examples include gas absorption into a liquid or liquid evaporation into a gas phase
• Chemical reactions can lead to the production or consumption of species in multiphase systems
  • Homogeneous reactions occur within a single phase, while heterogeneous reactions take place at the interface between phases
• Mass transfer limitations can significantly affect the overall performance of multiphase systems
  • Examples include gas-liquid reactions limited by the diffusion of reactants across the interface
• Enhancement factors are used to account for the increase in mass transfer due to chemical reactions or other mechanisms

Interfacial Phenomena

• Surface tension is the force that acts along the interface between two immiscible fluids
  • Arises from the imbalance of molecular forces at the interface and tends to minimize the interfacial area
• Capillary pressure is the pressure difference across a curved interface due to surface tension
  • Described by the Young-Laplace equation, which relates capillary pressure to surface tension and interface curvature
• Wettability refers to the tendency of a fluid to spread on or adhere to a solid surface in the presence of another immiscible fluid
  • Characterized by the contact angle formed between the fluid-fluid interface and the solid surface
• Marangoni effect is the mass transfer along an interface due to surface tension gradients
  • Can be induced by temperature or concentration gradients along the interface
• Interfacial instabilities (Rayleigh-Taylor, Kelvin-Helmholtz) can lead to the breakup or deformation of interfaces
  • Governed by the balance between stabilizing and destabilizing forces, such as surface tension and density differences
• Coalescence is the merging of two or more dispersed phase elements (bubbles or droplets) into a single larger element
• Breakup is the fragmentation of a dispersed phase element into smaller elements due to hydrodynamic forces or instabilities

Numerical Methods and Simulations

• Finite difference methods discretize the governing equations using a grid of points and approximate derivatives with finite differences
  • Explicit schemes calculate the solution at the current time step using information from the previous time step
  • Implicit schemes solve a system of equations involving both the current and previous time steps
• Finite volume methods divide the computational domain into control volumes and solve the conservation equations in integral form
  • Fluxes across control volume faces are evaluated using interpolation schemes (upwind, central differencing)
• Finite element methods approximate the solution using a weighted sum of basis functions defined on a mesh of elements
  • Particularly suitable for complex geometries and adaptive mesh refinement
• Lagrangian-Eulerian methods track the dispersed phase elements individually (Lagrangian) while solving the continuous phase equations on a fixed grid (Eulerian)
  • Examples include the Discrete Element Method (DEM) for particle-laden flows and the Discrete Bubble Model (DBM) for bubbly flows
• Interface tracking methods explicitly track the position and shape of the interface between different phases
  • Examples include the Volume of Fluid (VOF) method and the Level Set method
• Turbulence modeling is essential for accurately simulating multiphase flows with significant turbulent fluctuations
  • Reynolds-Averaged Navier-Stokes (RANS) models (k-ε, k-ω) provide averaged descriptions of turbulence
  • Large Eddy Simulation (LES) resolves large-scale turbulent structures while modeling the smaller scales

Applications and Case Studies

• Fluidized beds are widely used in chemical and process industries for gas-solid reactions, drying, and mixing
  • Solid particles are suspended by an upward flow of fluid, creating a fluidized state with enhanced heat and mass transfer
• Bubble columns are reactor vessels in which gas is dispersed as bubbles in a continuous liquid phase
  • Used for gas-liquid reactions, fermentation processes, and wastewater treatment
• Spray drying is a process where a liquid feed is atomized into droplets and dried by hot gas to produce a powder
  • Applications include food, pharmaceutical, and ceramic industries
• Multiphase flow in pipelines is encountered in oil and gas production, where oil, gas, and water are transported together
  • Flow regimes (stratified, slug, annular) and pressure drop prediction are important considerations
• Sediment transport in rivers and coastal environments involves the movement of solid particles by flowing water
  • Understanding erosion, deposition, and morphological changes is crucial for river management and coastal protection
• Fuel injection in internal combustion engines involves the atomization and mixing of liquid fuel with air
  • Multiphase flow modeling is used to optimize injection strategies and reduce emissions
• Boiling and condensation processes involve phase change heat transfer and are critical in power generation and refrigeration systems
  • Nucleate boiling, film boiling, and dropwise condensation are examples of multiphase flow regimes
• Blood flow in the human body is a complex multiphase flow problem involving red blood cells, white blood cells, and plasma
  • Modeling blood flow is important for understanding cardiovascular diseases and designing medical devices