The Mixture Model Equations
In the Mixture Model interfaces the particle-fluid combination is regarded as a single flowing continuum with macroscopic properties such as density and viscosity. The two phases consist of one dispersed phase and one continuous phase. The mixture model is valid if the continuous phase is a liquid, and the dispersed phase consists of solid particles, liquid droplets, or gas bubbles. For gas bubbles in a liquid, however, the bubbly flow model is preferable. The mixture model relies on the following assumptions:
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The density of each phase is approximately constant.
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Both phases share the same pressure field.
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The particle relaxation time is short compared to the time-scales of the macroscopic flow.
The mixture density is given by
where
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and denote the volume fractions of the continuous phase and the dispersed phase (SI unit: m3/m3), respectively
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ρc is the continuous phase density (SI unit: kg/m3), and
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ρd is the dispersed phase density (SI unit: kg/m3).
The volume flux for each phase is
where uc and ud (SI unit: m/s) are the continuous and the dispersed phase velocity vectors, respectively.
In previous versions of COMSOL Multiphysics (prior to version 5.4, see Mass-Averaged Mixture Velocity), the mixture velocity used was the mass-averaged mixture velocity u (SI unit: m/s), defined as
(6-41)
The mixture velocity used here is the volume-averaged flux density, or volume-averaged mixture velocity j (SI unit: m/s), defined as
The continuity equation for the mixture is
(6-42)
In the Mixture Model interfaces it is assumed that the densities of both phases, ρc and ρd, are constant, and therefore the following alternative form of the continuity equation for the mixture is used
(6-43)
The momentum equation for the mixture is
where:
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j is the velocity vector (SI unit: m/s)
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ρ is the density (SI unit: kg/m3)
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p is the pressure (SI unit: Pa)
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ε is the reduced density difference (SI unit: kg/kg)
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uslip is the slip velocity vector between the two phases (SI unit: m/s)
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jslip is the slip flux (SI unit: m/s)
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τGm is the sum of the viscous and turbulent stresses (SI unit: kg/(m·s2))
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Dmd is a turbulent dispersion coefficient (SI unit: m2/s)
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mdc is the mass transfer rate from the dispersed to the continuous phase (SI unit: kg/(m3·s))
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g is the gravity vector (SI unit: m/s2), and
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F is any additional volume force (SI unit: N/m3)
The slip flux is defined for convenience as,
(6-44)
Here, uslip (SI unit: m/s) denotes the relative velocity between the two phases. For different available models for the slip velocity, see Slip Velocity Models.
The reduced density difference ε is given by
The sum of the viscous and turbulent stresses is
(6-45)
where μ (SI unit: Pa·s) is the mixture viscosity, μT (SI unit: Pa·s) the turbulent viscosity, and k (SI unit: m2/s2) is the turbulent kinetic energy, when available. If no turbulence model is used, μT and k equal zero
The transport equation for , the dispersed phase volume fraction, is
(6-46)
where mdc (SI unit: kg/(m3·s)) is the mass transfer rate from the dispersed to the continuous phase and Dmd is a turbulent dispersion coefficient (SI unit: m2/s) (see Turbulence Modeling in Mixture Models), accounting for extra diffusion due to turbulent eddies. When a turbulence model is not used, Dmd is zero. Assuming constant density for the dispersed phase and using Equation 6-43, Equation 6-46 can be rewritten as
(6-47)
The continuous phase volume fraction is
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 dispersed phase to the continuous mdc (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 dispersed phase number density (that is, the number of particles per volume) in addition to the phase volume fraction. The Mixture Model interface assumes that the particles can increase or decrease in size 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 dispersed particles give the interfacial area per unit volume (SI unit: m2/m3):
Mass-Averaged Mixture Velocity
In previous versions of COMSOL Multiphysics (prior to version 5.4), the mixture velocity used was the mass-averaged mixture velocity u (SI unit: m/s), see Equation 6-41. The continuity equation was
(6-48)
and the transport equation for d was
(6-49)
The dispersed phase velocity vector ud and the mass fraction of the dispersed phase cd are given by
The continuity equation Equation 6-48 is incompatible with the no-penetration condition of u at the walls, and u is not divergence-free. The volume-averaged velocity j is solenoidal and Equation 6-43 is compatible with the no-penetration condition at the walls (as long as there is no mass transfer or the densities of the two phases are equal).