💨Mathematical Fluid Dynamics Unit 10 – Multiphase & Multicomponent Flows

Key Concepts and Definitions

• Multiphase flow involves the simultaneous presence of two or more phases (gas, liquid, or solid) in a system
• Multicomponent flow consists of a mixture of different chemical species or components within a single phase
• Phase refers to a distinct state of matter (gas, liquid, or solid) with homogeneous physical properties
• Interface represents the boundary between two phases where properties change discontinuously
• Volume fraction $(\alpha_k)$ quantifies the proportion of each phase $k$ in a given volume
  • Defined as the ratio of the volume occupied by phase $k$ to the total volume
  • Satisfies the constraint $\sum_{k=1}^N \alpha_k = 1$, where $N$ is the total number of phases
• Interfacial area concentration $(\alpha_i)$ measures the amount of interfacial area per unit volume
• Slip velocity $(\vec{v}_{slip})$ represents the relative velocity between phases
  • Calculated as the difference between the phase velocities $\vec{v}_{slip} = \vec{v}_1 - \vec{v}_2$
• Interfacial transfer terms describe the exchange of mass, momentum, and energy between phases

Fundamental Equations

• Conservation of mass (continuity equation) for each phase $k$:
  • $\frac{\partial (\alpha_k \rho_k)}{\partial t} + \nabla \cdot (\alpha_k \rho_k \vec{v}_k) = \Gamma_k$
  • $\rho_k$ is the density of phase $k$, $\vec{v}_k$ is the velocity of phase $k$, and $\Gamma_k$ represents mass transfer to phase $k$
• Conservation of momentum for each phase $k$:
  • $\frac{\partial (\alpha_k \rho_k \vec{v}_k)}{\partial t} + \nabla \cdot (\alpha_k \rho_k \vec{v}_k \vec{v}_k) = -\alpha_k \nabla p + \nabla \cdot (\alpha_k \tau_k) + \alpha_k \rho_k \vec{g} + \vec{M}_k$
  • $p$ is the pressure, $\tau_k$ is the stress tensor of phase $k$, $\vec{g}$ is the gravitational acceleration, and $\vec{M}_k$ represents interfacial momentum transfer to phase $k$
• Conservation of energy for each phase $k$:
  • $\frac{\partial (\alpha_k \rho_k E_k)}{\partial t} + \nabla \cdot (\alpha_k \rho_k H_k \vec{v}_k) = -\nabla \cdot (\alpha_k \vec{q}_k) + \alpha_k \frac{Dp}{Dt} + \nabla \cdot (\alpha_k \tau_k \cdot \vec{v}_k) + \alpha_k \rho_k \vec{g} \cdot \vec{v}_k + Q_k$
  • $E_k$ is the total energy of phase $k$, $H_k$ is the total enthalpy of phase $k$, $\vec{q}_k$ is the heat flux vector of phase $k$, and $Q_k$ represents interfacial energy transfer to phase $k$
• Closure relations are needed to model interfacial transfer terms, phase interactions, and constitutive equations

Classification of Multiphase Flows

• Gas-liquid flows (bubbly flow, slug flow, churn flow, annular flow)
• Gas-solid flows (pneumatic conveying, fluidized beds)
• Liquid-liquid flows (immiscible liquids, emulsions)
• Liquid-solid flows (slurry transport, sedimentation)
• Three-phase flows (gas-liquid-solid systems)
• Classified based on the state of the dispersed phase
  • Bubbly flows have gas bubbles dispersed in a continuous liquid phase
  • Droplet flows have liquid droplets dispersed in a continuous gas phase
  • Particulate flows have solid particles dispersed in a continuous gas or liquid phase
• Flow patterns depend on factors such as phase properties, flow rates, and geometry
  • Horizontal pipes exhibit stratified, wavy, slug, and annular flow patterns
  • Vertical pipes display bubbly, slug, churn, and annular flow patterns

Interfacial Phenomena

• Surface tension $(\sigma)$ arises from the imbalance of molecular forces at the interface
  • Causes the interface to minimize its surface area and form spherical shapes (bubbles, droplets)
  • Quantified as the force per unit length acting tangentially to the interface
• Capillary pressure $(\Delta p_c)$ is the pressure difference across a curved interface
  • Related to surface tension and interface curvature by the Young-Laplace equation: $\Delta p_c = \sigma (\frac{1}{R_1} + \frac{1}{R_2})$
  • $R_1$ and $R_2$ are the principal radii of curvature
• Marangoni effect refers to the mass transfer along an interface due to surface tension gradients
  • Caused by temperature or concentration gradients
  • Induces interfacial flow from regions of low surface tension to high surface tension
• Wetting and contact angles describe the interaction between a liquid and a solid surface
  • Contact angle $(\theta)$ is the angle formed by the liquid-vapor interface and the solid surface
  • Wetting occurs when $\theta < 90^\circ$, non-wetting when $\theta > 90^\circ$
• Interfacial instabilities (Rayleigh-Taylor, Kelvin-Helmholtz) can lead to the breakup of interfaces
  • Rayleigh-Taylor instability occurs when a denser fluid is above a lighter fluid
  • Kelvin-Helmholtz instability arises from velocity shear at the interface between two fluids

Flow Regimes and Patterns

• Flow regimes characterize the spatial distribution of phases in a multiphase system
• Determined by factors such as phase properties, flow rates, and geometry
• Gas-liquid flow regimes in vertical pipes:
  • Bubbly flow: discrete gas bubbles dispersed in a continuous liquid phase
  • Slug flow: large bullet-shaped gas bubbles (Taylor bubbles) separated by liquid slugs
  • Churn flow: chaotic and oscillatory flow with irregular gas structures
  • Annular flow: gas flows in the core, and liquid flows as a film along the pipe wall
• Gas-liquid flow regimes in horizontal pipes:
  • Stratified flow: gas and liquid flow separately with a smooth interface
  • Wavy flow: gas and liquid flow separately with a wavy interface
  • Slug flow: intermittent liquid slugs and elongated gas bubbles
  • Annular flow: similar to vertical annular flow
• Flow pattern maps (Baker chart, Taitel-Dukler map) predict flow regimes based on dimensionless parameters
  • Dimensionless parameters include superficial velocities, density ratio, and viscosity ratio
• Transition criteria between flow regimes based on stability analysis and empirical correlations

Modeling Techniques

• Eulerian-Eulerian (two-fluid) model treats each phase as a separate continuum
  • Solves averaged conservation equations for each phase
  • Requires closure relations for interfacial transfer terms and phase interactions
  • Suitable for dispersed flows with high volume fractions
• Eulerian-Lagrangian model treats the continuous phase as a continuum and the dispersed phase as discrete particles
  • Solves averaged conservation equations for the continuous phase
  • Tracks individual particles using Newton's second law of motion
  • Accounts for particle-fluid and particle-particle interactions
  • Suitable for dilute dispersed flows with low volume fractions
• Mixture model considers the multiphase flow as a single fluid with averaged properties
  • Solves conservation equations for the mixture
  • Uses algebraic expressions to determine phase velocities and volume fractions
  • Applicable when the phases are strongly coupled and have similar velocities
• Interface tracking methods (Volume of Fluid, Level Set) explicitly capture the interface between phases
  • Volume of Fluid (VOF) method uses a color function to represent the volume fraction of each phase in a cell
  • Level Set method defines the interface as a zero level set of a signed distance function
  • Accurately resolve interface geometry and topology changes
• Averaging techniques (time averaging, volume averaging, ensemble averaging) derive macroscopic equations from microscopic descriptions
  • Time averaging is suitable for steady flows
  • Volume averaging is used for systems with spatial inhomogeneities
  • Ensemble averaging is employed for turbulent and stochastic flows

Numerical Methods and Simulations

• Finite volume method (FVM) is widely used for multiphase flow simulations
  • Discretizes the computational domain into control volumes
  • Enforces conservation principles by balancing fluxes across control volume faces
  • Handles complex geometries and unstructured grids
• Staggered grid arrangement stores velocity components at cell faces and scalar quantities at cell centers
  • Prevents checkerboard pressure oscillations
  • Facilitates the implementation of momentum interpolation schemes (QUICK, MUSCL)
• Pressure-velocity coupling algorithms (SIMPLE, PISO) solve the coupled system of equations
  • SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) is an iterative algorithm
  • PISO (Pressure Implicit with Splitting of Operators) is a non-iterative algorithm
• Advection schemes (upwind, central differencing, TVD) discretize the convective terms in the governing equations
  • Upwind schemes are stable but prone to numerical diffusion
  • Central differencing schemes are second-order accurate but may introduce oscillations
  • TVD (Total Variation Diminishing) schemes combine the advantages of upwind and central differencing
• Temporal discretization schemes (explicit, implicit, Crank-Nicolson) integrate the equations in time
  • Explicit schemes are straightforward to implement but have stability restrictions
  • Implicit schemes are unconditionally stable but require the solution of a linear system
  • Crank-Nicolson scheme is second-order accurate and unconditionally stable
• Turbulence modeling (RANS, LES, DNS) captures the effects of turbulence on multiphase flows
  • RANS (Reynolds-Averaged Navier-Stokes) models solve for averaged quantities and model turbulence effects
  • LES (Large Eddy Simulation) resolves large-scale turbulent structures and models sub-grid scale effects
  • DNS (Direct Numerical Simulation) resolves all scales of turbulence but is computationally expensive

Applications and Case Studies

• Oil and gas industry: multiphase flow in pipelines, separators, and wells
  • Predicting flow patterns and pressure drops in pipelines
  • Designing efficient separators for oil-water-gas mixtures
  • Modeling gas-liquid flow in wellbores and risers
• Chemical and process engineering: reactors, heat exchangers, and distillation columns
  • Simulating gas-liquid reactors (bubble columns, airlift reactors) for mass transfer and reaction kinetics
  • Designing multi-phase heat exchangers for enhanced heat transfer
  • Modeling distillation columns for separation processes
• Nuclear engineering: two-phase flow in boiling water reactors (BWRs) and pressurized water reactors (PWRs)
  • Predicting void fraction and critical heat flux in BWRs
  • Analyzing departure from nucleate boiling (DNB) in PWRs
  • Simulating two-phase flow instabilities and flow-induced vibrations
• Environmental engineering: sediment transport, bubble plumes, and oil spills
  • Modeling sediment transport in rivers and coastal areas
  • Simulating bubble plumes for aeration and mixing in water bodies
  • Predicting the fate and transport of oil spills in marine environments
• Biomedical engineering: blood flow, drug delivery, and microfluidic devices
  • Simulating blood flow as a multiphase mixture of plasma and blood cells
  • Modeling drug delivery systems with nanoparticles or microbubbles
  • Designing microfluidic devices for cell separation and analysis
• Aerospace engineering: fuel injection, spray combustion, and icing
  • Simulating fuel injection and atomization in combustion chambers
  • Modeling spray combustion and pollutant formation
  • Predicting ice accretion on aircraft wings and engines
• Geophysical flows: volcanic eruptions, geysers, and hydrothermal vents
  • Simulating gas-particle flows in volcanic plumes and pyroclastic density currents
  • Modeling geyser eruptions driven by two-phase flow in geothermal systems
  • Investigating multiphase flow in hydrothermal vents and black smokers