Slip Velocity Models
The Mixture Model interfaces contain three predefined models for the relative velocity between the two phases uslip (SI unit: m/s):
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The Schiller-Naumann model
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The Haider-Levenspiel model
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The Hadamard-Rybczynski model
All three models use the following relation for the slip velocity:
(6-50)
where Cd (dimensionless) is the particle drag coefficient. Essentially, interpret the relation as a balance between viscous drag and buoyancy forces acting on the dispersed phase.
The Schiller-Naumann model models the drag coefficient according to
where Rep is the particle Reynolds number
Note that this definition of the particle Reynolds number uses the viscosity of the mixture and not the viscosity of the continuous phase, as would be customary for a single particle in a pure fluid with viscosity μc. This choice incorporates the hindrance effect of the other particles on the slip velocity, see Ref. 4.
Because the particle Reynolds number depends on the slip velocity, an implicit equation must be solved to obtain the slip velocity. Therefore, the Mixture Model interfaces add an additional equation for
when the Schiller-Naumann slip model is used. The Schiller-Naumann model is particularly well-suited for solid particles in a liquid.
The Haider-Levenspiel model is applicable to nonspherical particles. It models the drag coefficient according to
where A, B, C, and D are empirical correlations of the particle sphericity. The sphericity is defined as the ratio of the surface area of a volume equivalent sphere to the surface area of the considered nonspherical particle
The correlation coefficients are given by
The diameter used in the particle Reynolds number is that of the volume equivalent sphere. The equation for the squared slip velocity is also added when the Haider-Levenspiel slip model is used.
The Hadamard-Rybczynski drag law is valid for particle Reynolds numbers less than 1, for particles, bubbles, and droplets. The drag coefficient for liquid droplets or bubbles is
which yields the following explicit expression for the slip velocity
For solid particles, the slip velocity is given by
when Rep < 1. For very small gas bubbles, the drag coefficient is observed to be closer to the solid-particle value. This is believed to be caused by surface-active impurities collecting on the bubble surface.