2.5 Drift-flux model

The drift-flux model simplifies multiphase flow modeling by treating the mixture as a whole, rather than separate phases. It introduces drift velocity to account for relative motion between phases, striking a balance between simplicity and accuracy for various flow regimes.

This model is widely used in petroleum, nuclear, and chemical industries for predicting key flow parameters. It's particularly useful for vertical and inclined pipes, where buoyancy and slip effects are significant, but can also be applied to horizontal flows with modifications.

Drift-flux model overview

• The drift-flux model is a simplified approach to modeling multiphase flows, particularly gas-liquid flows, by considering the mixture as a whole rather than treating each phase separately
• Assumes that the phases are interpenetrating continua and introduces the concept of drift velocity to account for the relative motion between the phases
• Provides a balance between the simplicity of the homogeneous model and the complexity of separated flow models, making it suitable for a wide range of applications in multiphase flow modeling

Assumptions and limitations

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• Assumes that the phases are well-mixed and have the same pressure at any given cross-section of the flow
• Neglects the detailed interface structure and the relative motion between the phases on a small scale
• Limited to dispersed bubble flow, slug flow, and churn flow regimes, where the assumptions of interpenetrating continua and local equilibrium are valid
• May not accurately capture the behavior of flows with significant phase separation or slip, such as annular flow or stratified flow

Applicability and flow regimes

• Widely used in the petroleum industry for modeling multiphase flow in wells, pipelines, and risers
• Suitable for vertical, inclined, and horizontal pipes, as well as annular and slug flow regimes
• Can be applied to a variety of gas-liquid systems, including air-water, steam-water, and oil-gas mixtures
• Provides reasonable predictions of void fraction, pressure gradient, and flow pattern transitions in many practical situations

Drift-flux model derivation

• The drift-flux model is derived from the fundamental conservation equations for mass and momentum, along with constitutive relationships and closure laws
• Introduces the concept of drift velocity to account for the relative motion between the phases and the distribution parameter to characterize the non-uniform velocity and concentration profiles

Conservation equations

• Mass conservation equation for the mixture: $\frac{\partial \rho_m}{\partial t} + \nabla \cdot (\rho_m \vec{v}_m) = 0$
  • $\rho_m$ is the mixture density
  • $\vec{v}_m$ is the mixture velocity
• Momentum conservation equation for the mixture: $\frac{\partial (\rho_m \vec{v}_m)}{\partial t} + \nabla \cdot (\rho_m \vec{v}_m \vec{v}_m) = -\nabla P + \rho_m \vec{g} + \nabla \cdot \overline{\overline{\tau}}_m$
  • $P$ is the pressure
  • $\vec{g}$ is the gravitational acceleration
  • $\overline{\overline{\tau}}_m$ is the mixture stress tensor

Constitutive relationships

• Mixture density: $\rho_m = \alpha \rho_g + (1 - \alpha) \rho_l$
  • $\alpha$ is the void fraction
  • $\rho_g$ and $\rho_l$ are the gas and liquid densities, respectively
• Mixture velocity: $\vec{v}_m = \frac{\alpha \rho_g \vec{v}_g + (1 - \alpha) \rho_l \vec{v}_l}{\rho_m}$
  • $\vec{v}_g$ and $\vec{v}_l$ are the gas and liquid velocities, respectively

Closure laws and correlations

• Drift velocity: $\vec{v}_{gj} = \vec{v}_g - \vec{v}_m = f(\alpha, \rho_g, \rho_l, \mu_g, \mu_l, \sigma, d_p, \vec{g})$
  • $\vec{v}_{gj}$ is the drift velocity
  • $\mu_g$ and $\mu_l$ are the gas and liquid viscosities, respectively
  • $\sigma$ is the surface tension
  • $d_p$ is the characteristic particle or bubble diameter
• Distribution parameter: $C_0 = f(\alpha, \text{flow regime}, \text{pipe geometry})$
  • $C_0$ is the distribution parameter, which accounts for the non-uniform velocity and concentration profiles

Drift velocity

• Drift velocity represents the relative velocity between the gas phase and the mixture, capturing the effects of buoyancy, drag, and other interfacial forces
• Plays a crucial role in the drift-flux model by accounting for the slip between the phases and the non-uniform velocity profiles

Definition and physical meaning

• Drift velocity is defined as the difference between the gas velocity and the mixture velocity: $\vec{v}_{gj} = \vec{v}_g - \vec{v}_m$
• Represents the velocity at which the gas phase "drifts" or moves relative to the mixture
• Positive drift velocity indicates that the gas phase is moving faster than the mixture, while negative drift velocity suggests that the gas phase is lagging behind the mixture

Factors affecting drift velocity

• Void fraction: Higher void fractions generally lead to higher drift velocities due to increased buoyancy effects
• Fluid properties: Density, viscosity, and surface tension of the gas and liquid phases influence the drift velocity through their effects on bubble rise velocity and drag forces
• Flow regime: Drift velocity varies depending on the flow regime (bubbly, slug, churn, annular), as the interfacial structure and relative motion between the phases change
• Pipe inclination: Drift velocity is affected by the inclination angle of the pipe, with higher drift velocities in vertical upward flows compared to horizontal or downward flows

Drift velocity correlations

• Several empirical correlations have been developed to estimate the drift velocity based on fluid properties, flow conditions, and pipe geometry
• Zuber-Findlay correlation: $v_{gj} = 1.53 \left[\frac{g \sigma (\rho_l - \rho_g)}{\rho_l^2}\right]^{1/4}$ for bubbly and slug flows in vertical pipes
  • $g$ is the gravitational acceleration
  • $\sigma$ is the surface tension
  • $\rho_l$ and $\rho_g$ are the liquid and gas densities, respectively
• Hasan-Kabir correlation: $v_{gj} = 0.75 \left[\frac{g (\rho_l - \rho_g) d_p}{\rho_l}\right]^{1/2}$ for bubbly flow in vertical and inclined pipes
  • $d_p$ is the bubble diameter
• Other correlations, such as the Ishii-Chawla, Kataoka-Ishii, and Hibiki-Ishii correlations, have been proposed for various flow conditions and pipe geometries

Distribution parameter

• The distribution parameter accounts for the non-uniform velocity and concentration profiles in multiphase flows, reflecting the effects of phase slip and flow pattern on the cross-sectional distribution of the phases
• Plays a crucial role in the drift-flux model by relating the average void fraction to the local void fraction and velocity profiles

Definition and physical interpretation

• The distribution parameter is defined as the ratio of the average void fraction to the volumetric quality: $C_0 = \frac{\langle \alpha \rangle}{\langle \alpha \rangle_{v}}$
  • $\langle \alpha \rangle$ is the average void fraction
  • $\langle \alpha \rangle_{v}$ is the volumetric quality, defined as $\frac{Q_g}{Q_g + Q_l}$, where $Q_g$ and $Q_l$ are the gas and liquid volumetric flow rates, respectively
• Represents the degree of non-uniformity in the velocity and concentration profiles across the pipe cross-section
• A value of $C_0 = 1$ indicates a uniform distribution, while $C_0 > 1$ suggests a higher concentration of the gas phase in the center of the pipe, and $C_0 < 1$ implies a higher concentration near the pipe wall

Factors influencing distribution parameter

• Flow regime: The distribution parameter varies significantly with the flow regime, as the phase distribution and velocity profiles change from bubbly to slug, churn, and annular flows
• Pipe geometry: The shape and size of the pipe cross-section affect the distribution parameter, with larger values observed in round pipes compared to rectangular or triangular channels
• Void fraction: The distribution parameter generally increases with increasing void fraction, as the gas phase tends to concentrate in the center of the pipe at higher void fractions
• Fluid properties: The density, viscosity, and surface tension of the phases influence the distribution parameter through their effects on bubble size, shape, and distribution

Distribution parameter correlations

• Several empirical correlations have been proposed to estimate the distribution parameter based on flow conditions, pipe geometry, and fluid properties
• Zuber-Findlay correlation: $C_0 = 1.2$ for bubbly and slug flows in vertical round pipes
• Ishii-Hibiki correlation: $C_0 = 1.2 - 0.2 \sqrt{\frac{\rho_g}{\rho_l}}$ for bubbly and slug flows in vertical round pipes
  • $\rho_g$ and $\rho_l$ are the gas and liquid densities, respectively
• Hibiki-Ishii correlation: $C_0 = 1.2 - 0.2 \sqrt{\frac{\rho_g}{\rho_l}} + 0.2 \left(\frac{D}{D_0}\right)^{0.5}$ for bubbly and slug flows in vertical round pipes
  • $D$ is the pipe diameter
  • $D_0$ is a reference diameter (typically 50 mm)
• Other correlations, such as the Kataoka-Ishii and Mishima-Hibiki correlations, have been developed for various flow conditions and pipe geometries

Void fraction prediction

• Void fraction is a key parameter in multiphase flow modeling, representing the fraction of the pipe cross-sectional area occupied by the gas phase
• The drift-flux model provides a means to predict the void fraction based on the mixture velocity, drift velocity, and distribution parameter

Void fraction calculation methods

• Drift-flux model expression for void fraction: $\alpha = \frac{j_g}{C_0 j_m + v_{gj}}$
  • $\alpha$ is the void fraction
  • $j_g$ is the superficial gas velocity
  • $j_m$ is the mixture velocity
  • $C_0$ is the distribution parameter
  • $v_{gj}$ is the drift velocity
• Iterative solution: The void fraction can be calculated iteratively using the drift-flux model expression, as the drift velocity and distribution parameter may depend on the void fraction itself
• Explicit correlations: Some explicit correlations have been proposed to estimate the void fraction directly from the superficial gas and liquid velocities, such as the Lockhart-Martinelli correlation and the Dix correlation

Void fraction vs mixture velocity

• The void fraction generally increases with increasing mixture velocity, as higher gas flow rates lead to a larger fraction of the pipe cross-section being occupied by the gas phase
• The relationship between void fraction and mixture velocity depends on the flow regime, with different trends observed in bubbly, slug, churn, and annular flows
• In bubbly and slug flows, the void fraction increases gradually with mixture velocity, while in churn and annular flows, the void fraction increases more rapidly and may approach unity at high mixture velocities

Void fraction vs gas and liquid velocities

• The void fraction depends on both the superficial gas velocity (j_g) and the superficial liquid velocity (j_l)
• Increasing the superficial gas velocity while keeping the superficial liquid velocity constant leads to an increase in void fraction
• Increasing the superficial liquid velocity while keeping the superficial gas velocity constant results in a decrease in void fraction
• The combined effect of gas and liquid velocities on void fraction can be visualized using flow pattern maps, which delineate the regions of different flow regimes based on the superficial velocities of the phases

Pressure gradient calculation

• Pressure gradient is an important parameter in multiphase flow design and analysis, as it determines the pumping power requirements and affects the flow behavior
• The drift-flux model allows for the calculation of the pressure gradient based on the contributions of friction, gravity, and acceleration

Pressure gradient components

• Total pressure gradient: $\left(\frac{dP}{dz}\right)_{total} = \left(\frac{dP}{dz}\right)_{friction} + \left(\frac{dP}{dz}\right)_{gravity} + \left(\frac{dP}{dz}\right)_{acceleration}$
  • $\left(\frac{dP}{dz}\right)_{friction}$ is the frictional pressure gradient
  • $\left(\frac{dP}{dz}\right)_{gravity}$ is the gravitational pressure gradient
  • $\left(\frac{dP}{dz}\right)_{acceleration}$ is the acceleration pressure gradient (usually negligible in steady-state flows)

Frictional pressure gradient

• Frictional pressure gradient accounts for the pressure drop due to the shear stress between the fluid and the pipe wall
• Can be estimated using the two-phase friction factor multiplier approach: $\left(\frac{dP}{dz}\right)_{friction} = \phi_l^2 \left(\frac{dP}{dz}\right)_{l}$
  • $\phi_l^2$ is the two-phase friction factor multiplier, which can be calculated using correlations such as the Lockhart-Martinelli or the Friedel correlation
  • $\left(\frac{dP}{dz}\right)_{l}$ is the single-phase liquid pressure gradient, calculated using the Darcy-Weisbach equation or the Haaland equation

Gravitational pressure gradient

• Gravitational pressure gradient represents the pressure change due to the elevation difference and the hydrostatic head of the mixture
• Calculated using the void fraction and the densities of the phases: $\left(\frac{dP}{dz}\right)_{gravity} = [\alpha \rho_g + (1 - \alpha) \rho_l] g \sin \theta$
  • $\alpha$ is the void fraction
  • $\rho_g$ and $\rho_l$ are the gas and liquid densities, respectively
  • $g$ is the gravitational acceleration
  • $\theta$ is the angle of inclination from the horizontal (positive for upward flow, negative for downward flow)

Drift-flux model applications

• The drift-flux model has been widely applied in various industries, including petroleum, nuclear, and chemical engineering, for the design and analysis of multiphase flow systems
• Provides a practical and computationally efficient approach to predict key flow parameters, such as void fraction, pressure gradient, and flow pattern transitions

Vertical and inclined pipes

• The drift-flux model is particularly suitable for modeling multiphase flow in vertical and inclined pipes, where the effects of buoyancy and slip are significant
• Has been successfully applied to predict void fraction, pressure gradient, and flow pattern transitions in oil and gas production wells, geothermal wells, and riser systems
• Provides a basis for the design and optimization of multiphase flow systems, such as gas lift operations, subsea pipelines, and heat exchangers

Horizontal and near-horizontal pipes

• The drift-flux model can also be applied to horizontal and near-horizontal pipes, although the effects of slip and phase stratification become more pronounced
• Modified drift-flux models have been proposed to account for the asymmetric phase distribution and the formation of stratified layers in horizontal flows
• Has been used to predict liquid holdup, pressure gradient, and flow pattern transitions in horizontal pipelines, such as in the transportation of oil-gas mixtures and in the design of multiphase flow metering systems

Annular and slug flow regimes

• The drift-flux model has been extended to model annular and slug flow regimes, which are commonly encountered in many industrial applications
• In annular flow, the model can be adapted to account for the presence of a liquid film on the pipe wall and the entrainment of liquid droplets in the gas core
• In slug flow, the model can be used to predict the characteristics of individual slugs, such as the slug length, frequency, and translational velocity, as well as the overall pressure gradient and void fraction
• Has been applied to the design and analysis of heat exchangers, condensers, and evaporators operating in annular and slug flow regimes

Drift-flux model extensions

• The basic drift-flux model has been extended and modified to account for various physical phenomena and flow conditions encountered in multiphase flow systems
• These extensions aim to improve the accuracy and applicability of the model in specific situations, such as in the presence of mass transfer, phase inversion, or transient flows

Incorporation of mass transfer

• Mass transfer between the phases, such as evaporation or condensation, can significantly affect the void fraction and pressure gradient in multiphase flows
• The drift-flux model has been extended to account for mass transfer by including source terms in the conservation equations and modifying the constitutive relationships
• Has been applied to model two-phase flow with phase change in steam generators, condensers, and evaporators, as well as in the modeling of gas-liquid reactions and absorption processes

Accounting for phase inversion

• Phase inversion refers to the transition from