1.3 Volume fraction and phase fraction

Volume fraction and phase fraction are key concepts in multiphase flow modeling. They quantify the spatial distribution and concentration of different phases within a system, allowing us to track how phases interact and evolve over time.

Understanding these fractions is crucial for accurately simulating multiphase flows. Volume fraction represents the space occupied by each phase, while phase fraction focuses on the mass distribution. Both play vital roles in determining mixture properties and flow behavior.

Volume fraction definition

• Volume fraction quantifies the volume occupied by a specific phase relative to the total volume of a multiphase system
• Represents the spatial distribution and local concentration of each phase within a control volume or computational cell
• Allows tracking the evolution and interaction of different phases in multiphase flow modeling

Ratio of phase volume to total volume

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• Mathematically expressed as the volume of a particular phase divided by the total volume of the mixture
• For phase $i$, the volume fraction is given by: $\alpha_i = \frac{V_i}{V_{\text{total}}}$
  • $V_i$ is the volume occupied by phase $i$
  • $V_{\text{total}}$ is the total volume of the mixture
• Volume fractions of all phases in a system sum up to unity: $\sum_{i=1}^{n} \alpha_i = 1$

Denoted by Greek letter alpha (α)

• Commonly represented by the Greek letter alpha ($\alpha$) in literature and equations
• Subscript $i$ is used to distinguish between different phases (e.g., $\alpha_g$ for gas phase, $\alpha_l$ for liquid phase)
• Volume fraction values range from 0 to 1, with 0 indicating the absence of a phase and 1 indicating a single-phase system

Phase fraction definition

• Phase fraction quantifies the mass of a specific phase relative to the total mass of a multiphase system
• Represents the mass distribution and composition of each phase within a control volume or computational cell
• Provides information about the mass-based concentration of different phases in multiphase flow modeling

Ratio of phase mass to total mass

• Mathematically expressed as the mass of a particular phase divided by the total mass of the mixture
• For phase $i$, the phase fraction is given by: $\beta_i = \frac{m_i}{m_{\text{total}}}$
  • $m_i$ is the mass of phase $i$
  • $m_{\text{total}}$ is the total mass of the mixture
• Phase fractions of all phases in a system sum up to unity: $\sum_{i=1}^{n} \beta_i = 1$

Denoted by Greek letter beta (β)

• Commonly represented by the Greek letter beta ($\beta$) in literature and equations
• Subscript $i$ is used to distinguish between different phases (e.g., $\beta_g$ for gas phase, $\beta_l$ for liquid phase)
• Phase fraction values range from 0 to 1, with 0 indicating the absence of a phase and 1 indicating a single-phase system

Relationship between volume and phase fractions

• Volume and phase fractions are related through the densities of individual phases and the mixture
• Understanding their relationship is crucial for converting between volume-based and mass-based quantities in multiphase flow modeling
• Enables consistent treatment of phase interactions and mixture properties

Derivation using phase and mixture densities

• The relationship between volume fraction ($\alpha_i$) and phase fraction ($\beta_i$) can be derived using phase densities ($\rho_i$) and mixture density ($\rho_m$)
• Starting from the definition of phase fraction: $\beta_i = \frac{m_i}{m_{\text{total}}} = \frac{\rho_i V_i}{\rho_m V_{\text{total}}}$
• Substituting the definition of volume fraction ($\alpha_i = \frac{V_i}{V_{\text{total}}}$) yields: $\beta_i = \frac{\rho_i \alpha_i}{\rho_m}$
• Mixture density can be expressed as a volume-weighted average of phase densities: $\rho_m = \sum_{i=1}^{n} \alpha_i \rho_i$

Simplification for incompressible flows

• For incompressible flows, where phase densities are constant, the relationship simplifies to: $\beta_i = \frac{\alpha_i \rho_i}{\sum_{j=1}^{n} \alpha_j \rho_j}$
• This simplification allows for easier conversion between volume and phase fractions in incompressible multiphase systems
• Commonly used in models for liquid-liquid or liquid-solid mixtures with constant phase densities

Volume fraction transport equation

• The volume fraction transport equation describes the spatial and temporal evolution of volume fractions in multiphase flow
• It accounts for the convective transport, diffusive transport, and source terms due to phase change and mass transfer
• Solving this equation is essential for tracking the distribution and interaction of phases in CFD simulations

Continuity equation for phase volume fraction

• The continuity equation for phase volume fraction is derived from the general continuity equation for each phase
• For phase $i$, the volume fraction transport equation is given by: $\frac{\partial \alpha_i}{\partial t} + \nabla \cdot (\alpha_i \mathbf{u}_i) = \frac{S_{\alpha_i}}{\rho_i}$
  • $\frac{\partial \alpha_i}{\partial t}$ represents the temporal change in volume fraction
  • $\nabla \cdot (\alpha_i \mathbf{u}_i)$ represents the convective transport term, where $\mathbf{u}_i$ is the velocity of phase $i$
  • $S_{\alpha_i}$ is the source term for volume fraction, accounting for phase change and mass transfer
  • $\rho_i$ is the density of phase $i$

Source terms for phase change and mass transfer

• The source term $S_{\alpha_i}$ in the volume fraction transport equation includes contributions from phase change and mass transfer
• Phase change (e.g., evaporation, condensation) can be modeled using appropriate phase change models, such as the Schrage equation or the Hertz-Knudsen equation
• Mass transfer between phases (e.g., dissolution, precipitation) can be modeled using mass transfer coefficients and concentration gradients
• The specific form of the source term depends on the physical processes and models used in the multiphase flow simulation

Phase fraction transport equation

• The phase fraction transport equation describes the spatial and temporal evolution of phase fractions in multiphase flow
• It accounts for the convective transport, diffusive transport, and source terms due to phase change and mass transfer
• Solving this equation is essential for tracking the composition and interaction of phases in CFD simulations

Continuity equation for phase mass fraction

• The continuity equation for phase mass fraction is derived from the general continuity equation for each phase, considering mass conservation
• For phase $i$, the phase fraction transport equation is given by: $\frac{\partial (\rho_m \beta_i)}{\partial t} + \nabla \cdot (\rho_m \beta_i \mathbf{u}_m) = S_{\beta_i}$
  • $\frac{\partial (\rho_m \beta_i)}{\partial t}$ represents the temporal change in phase mass fraction
  • $\nabla \cdot (\rho_m \beta_i \mathbf{u}_m)$ represents the convective transport term, where $\mathbf{u}_m$ is the mixture velocity
  • $S_{\beta_i}$ is the source term for phase fraction, accounting for phase change and mass transfer

Source terms for phase change and mass transfer

• The source term $S_{\beta_i}$ in the phase fraction transport equation includes contributions from phase change and mass transfer
• Similar to the volume fraction transport equation, phase change and mass transfer can be modeled using appropriate models and equations
• The specific form of the source term depends on the physical processes and models used in the multiphase flow simulation
• Consistency between the source terms in the volume fraction and phase fraction transport equations is important for mass conservation

Mixture properties using volume fraction

• Mixture properties, such as density, viscosity, and thermal conductivity, can be calculated using volume fractions and individual phase properties
• Volume fraction-based mixing rules are commonly used in multiphase flow modeling to determine the effective properties of the mixture
• These mixing rules assume that the mixture properties are weighted averages of the individual phase properties based on their volume fractions

Density

• The mixture density $\rho_m$ can be calculated as a volume-weighted average of the individual phase densities: $\rho_m = \sum_{i=1}^{n} \alpha_i \rho_i$
• This mixing rule assumes that the phases are well-mixed and have no volume change upon mixing
• It is widely used in multiphase flow models, such as the Eulerian-Eulerian approach

Viscosity

• The mixture viscosity $\mu_m$ can be estimated using various volume fraction-based mixing rules, such as:
  • Linear mixing rule: $\mu_m = \sum_{i=1}^{n} \alpha_i \mu_i$
  • Inverse mixing rule: $\frac{1}{\mu_m} = \sum_{i=1}^{n} \frac{\alpha_i}{\mu_i}$
• The choice of mixing rule depends on the specific multiphase system and the assumptions made about the interaction between phases

Thermal conductivity

• The mixture thermal conductivity $k_m$ can be calculated using volume fraction-based mixing rules, similar to density and viscosity
• Common mixing rules include:
  • Linear mixing rule: $k_m = \sum_{i=1}^{n} \alpha_i k_i$
  • Effective medium theory (EMT) mixing rule: $\sum_{i=1}^{n} \alpha_i \frac{k_i - k_m}{k_i + 2k_m} = 0$
• The appropriate mixing rule depends on the thermal properties and morphology of the multiphase system

Mixture properties using phase fraction

• Mixture properties can also be calculated using phase fractions and individual phase properties
• Phase fraction-based mixing rules are less common compared to volume fraction-based mixing rules but can be useful in certain situations
• These mixing rules assume that the mixture properties are weighted averages of the individual phase properties based on their mass fractions

Density

• The mixture density $\rho_m$ can be calculated as a mass-weighted average of the individual phase densities: $\frac{1}{\rho_m} = \sum_{i=1}^{n} \frac{\beta_i}{\rho_i}$
• This mixing rule is derived from the definition of phase fraction and the conservation of mass
• It is consistent with the volume fraction-based mixing rule for density

Viscosity

• The mixture viscosity $\mu_m$ can be estimated using phase fraction-based mixing rules, such as:
  • Mass-weighted mixing rule: $\mu_m = \sum_{i=1}^{n} \beta_i \mu_i$
  • Inverse mass-weighted mixing rule: $\frac{1}{\mu_m} = \sum_{i=1}^{n} \frac{\beta_i}{\mu_i}$
• The choice of mixing rule depends on the specific multiphase system and the assumptions made about the interaction between phases

Thermal conductivity

• The mixture thermal conductivity $k_m$ can be calculated using phase fraction-based mixing rules, similar to density and viscosity
• Common mixing rules include:
  • Mass-weighted mixing rule: $k_m = \sum_{i=1}^{n} \beta_i k_i$
  • Modified effective medium theory (EMT) mixing rule: $\sum_{i=1}^{n} \beta_i \frac{k_i - k_m}{k_i + 2k_m} = 0$
• The appropriate mixing rule depends on the thermal properties and morphology of the multiphase system

Volume vs phase fraction in CFD

• Volume fraction and phase fraction are both used in computational fluid dynamics (CFD) simulations of multiphase flows
• The choice between volume fraction and phase fraction depends on the specific multiphase flow model, numerical methods, and physical properties of the system
• Understanding the advantages and disadvantages of each approach is crucial for selecting the appropriate formulation for a given multiphase flow problem

Advantages and disadvantages

• Volume fraction formulation:
  • Advantages:
    • Directly relates to the spatial distribution of phases
    • Easier to visualize and interpret the phase distribution
    • Suitable for problems with significant volume changes (e.g., phase change, compressibility)
  • Disadvantages:
    • May require additional equations for phase densities if they are not constant
    • Can be more computationally expensive due to the need to solve volume fraction transport equations for each phase
• Phase fraction formulation:
  • Advantages:
    • Directly relates to the mass distribution of phases
    • Suitable for problems with mass transfer between phases
    • Can be more convenient when dealing with mass-based source terms and reactions
  • Disadvantages:
    • Less intuitive for visualizing the spatial distribution of phases
    • May require additional equations for phase densities if they are not constant
    • Can be less accurate for problems with significant volume changes

Numerical considerations

• Consistency between volume fraction and phase fraction formulations is important for mass conservation and accuracy
• Numerical schemes for solving volume fraction and phase fraction transport equations should be chosen carefully to ensure stability, accuracy, and boundedness
• Common numerical schemes include:
  • First-order upwind scheme
  • Second-order upwind scheme
  • High-resolution schemes (e.g., QUICK, MUSCL)
  • Flux-limiting schemes (e.g., TVD, NVD)
• Proper treatment of source terms and interphase coupling is crucial for maintaining consistency and avoiding numerical artifacts

Experimental measurement techniques

• Experimental measurement of volume fraction and phase fraction is essential for validating multiphase flow models and understanding the behavior of real multiphase systems
• Various techniques have been developed to measure volume fraction and phase fraction, each with its own advantages and limitations
• The choice of measurement technique depends on the specific multiphase system, flow conditions, and desired spatial and temporal resolution

Tomographic methods for volume fraction

• Tomographic methods provide non-invasive measurements of volume fraction distributions in multiphase flows
• Common tomographic techniques include:
  • Electrical capacitance tomography (ECT): Measures the permittivity distribution, which is related to the volume fraction distribution
  • Electrical resistance tomography (ERT): Measures the conductivity distribution, which is related to the volume fraction distribution
  • X-ray computed tomography (CT): Measures the attenuation of X-rays, which is related to the density and volume fraction distribution
  • Gamma-ray computed tomography (GCT): Measures the attenuation of gamma-rays, which is related to the density and volume fraction distribution
• Tomographic methods offer high spatial resolution but may have limited temporal resolution and require sophisticated data processing algorithms

Sampling methods for phase fraction

• Sampling methods involve extracting a representative sample of the multiphase mixture and measuring the phase fractions through various techniques
• Common sampling methods include:
  • Isokinetic sampling: Ensures that the sample is extracted at the same velocity as the main flow to avoid biasing the phase fractions
  • Grab sampling: Involves quickly capturing a sample of the multiphase mixture at a specific location and time
  • Separation techniques: Use physical or chemical methods to separate the phases in the sample, such as centrifugation, filtration, or solvent extraction
• Sampling methods provide direct measurements of phase fractions but are invasive and may disturb the flow field
• Proper sampling techniques and statistical analysis are crucial for obtaining reliable phase fraction measurements