The Bubbly Flow Equations (2024)

The Bubbly Flow Equations

The two-fluid Euler-Euler Model is a general, macroscopic model for two-phase fluid flow. It treats the two phases as interpenetrating media, tracking the averaged concentration of the phases. One velocity field is associated with each phase. A momentum balance equation and a continuity equation describe the dynamics of each of the phases. The bubbly flow model is a simplification of the two-fluid model, relying on the following assumptions:

•	The gas density is negligible compared to the liquid density.

•	The motion of the gas bubbles relative to the liquid is determined by a balance between viscous drag and pressure forces.

•	The two phases share the same pressure field.

Based on these assumptions, the momentum and continuity equations for the two phases can be combined and a gas phase transport equation is kept in order to track the volume fraction of the bubbles. The momentum equation is

(6-30)

In Equation6-30, the variables are as follows:

•	ul is the velocity vector (SI unit: m/s)

•	p is the pressure (SI unit: Pa)

•	is the phase volume fraction (SI unit: m3/m3)

•	ρ is the density (SI unit: kg/m3)

•	g is the gravity vector (SIunit:m/s2)

•	F is any additional volume force (SI unit: N/m3)

•	μl is the dynamic viscosity of the liquid (SI unit: Pa·s), and

•	μT is the turbulent viscosity (SI unit: Pa·s)

The subscripts “l” and “g” denote quantities related to the liquid phase and the gas phase, respectively.

The continuity equation is

(6-31)

and the gas phase transport equation is

(6-32)

where mgl is the mass transfer rate from the gas to the liquid (SI unit: kg/(m3·s)).

For low gas volume fractions (), you can replace the momentum equations, Equation6-30, and the continuity equation, Equation6-31, by

(6-33)

(6-34)

By default, the Laminar Bubbly Flow interface uses Equation6-33 and 6-34. To switch to Equation6-30 and 6-31, click to clear the Low gas concentration check box under the Physical Model section.

The physics interface solves for ul, p, and

the effective gas density. The gas velocity ug is the sum of the following velocities:

(6-35)

where uslip is the relative velocity between the phases and udrift is a drift velocity (see Turbulence Modeling in Bubbly Flow Applications). The physics interface calculates the gas density from the ideal gas law:

where M is the molecular weight of the gas (SI unit: kg/mol), R is the ideal gas constant (8.314472J/(mol·K)), pref a reference pressure (SI unit: Pa), and T is temperature (SIunit: K). pref is a scalar variable, which by default is 1atm (1 atmosphere or 101,325Pa). The liquid volume fraction is calculated from

When there is a drift velocity, it has the form

(6-36)

where

Here, μT is the turbulent viscosity, and σT is the turbulent Schmidt number.

Inserting Equation6-36 and Equation6-35 into Equation6-32 gives

That is, the drift velocity introduces a diffusive term in the gas transport equation. This is how the equation for the transport of the volume fraction of gas is actually implemented in the physics interface.

The bubbly-flow equation formulation is relatively simple, but it can display some nonphysical behavior. One is artificial accumulation of bubbles, for example, beneath walls where the pressure gradient forces the bubbles upward, but the bubbles have no place to go and there is no term in the model to prevent the volume fraction of gas from growing. To prevent this from happening, is set to μl in the laminar case. The only apparent effect of this in most cases where the bubbly-flow equations are applicable is that nonphysical accumulation of bubbles is reduced. The small effective viscosity in the transport equation for has beneficial effects on the numerical properties of the equation system.

Mass Transfer and Interfacial Area

It is possible to account for mass transfer between the two phases by specifying an expression for the mass transfer rate from the gas to the liquid mgl (SI unit: kg/(m3·s)).

The mass transfer rate typically depends on the interfacial area between the two phases. An example is when gas dissolves into a liquid. In order to determine the interfacial area, it is necessary to solve for the bubble number density (that is, the number of bubbles per volume) in addition to the phase volume fraction. The Bubbly Flow interface assumes that the gas bubbles can expand or shrink but not completely vanish, merge, or split. The conservation of the number density n (SI unit: 1/m3) then gives

The number density and the volume fraction of gas give the interfacial area per unit volume (SI unit: m2/m3):

Slip Model

The simplest possible approximation for the slip velocity uslip is to assume that the bubbles always follow the liquid phase; that is, uslip=0. This is known as hom*ogeneous flow.

A better model can be obtained from the momentum equation for the gas phase. Comparing size of different terms, it can be argued that the equation can be reduced to a balance between the viscous drag force, fD and the pressure gradient (Ref.3), a so called pressure-drag balance:

(6-37)

Here fD can be written as

(6-38)

where in turn db (SI unit: m) is the bubble diameter, and Cd (dimensionless) is the viscous drag coefficient. Given Cd and db, Equation6-37 can be used to calculate the slip velocity. In practice, Equation6-37 is multiplied by to reduce the slip velocity for large values of .

Schwarz and Turner (Ref.4) proposed a linearized version of Equation6-38 appropriate for air bubbles of 1–10 mm mean diameter in water:

(6-39)

The Hadamard-Rybczynski model is appropriate for small spherical bubbles with diameter less than 2mm, and bubble Reynolds number less than one. The model uses the following expression for the drag coefficient (Ref.5):

For bubbles with diameter larger than 2mm, the model suggested by Sokolichin, Eigenberger, and Lapin (Ref.1) is a more appropriate choice:

where Eö is the Eötvös number

Here, g is the magnitude of the gravity vector and σ the surface tension coefficient.