M A S S T R A N S F E R A N D I N T E R F A C I A L A R E A

It is possible to account for mass transfer between the two phases by specifying an expression for the mass transfer rate from gas to 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 the 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

nt + nug = 0

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

a = 4n 1 3 3 g 2 3

Turbulence Modeling in Bubbly Flow Applications

For most bubbly flow applications the flow field becomes turbulent. In that case, use a turbulence model and solve for the averaged velocity field. In The Turbulent Bubbly Flow Interface, use the k- turbulence model. In addition to the options of the single phase model, it is also possible to account for bubble-induced turbulence, that is, extra production of turbulence due to the relative motion between the gas bubbles and the liquid.

The k- model solves two extra transport equations for two additional variables: the turbulent kinetic energy, k (m2/s2) and the dissipation rate of turbulent energy,

(m/s3). The turbulent viscosity is then

where C is a model constant.

The transport equation for the turbulent kinetic energy, k, is

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where the production term is

and the evolution of the turbulent energy’s dissipation rate :

In all the previous equations, the velocity, ul, is the liquid phase averaged velocity field.

The standard k- turbulence model uses the following constants:

The included source term Sk accounts for bubble-induced turbulence and is given by

Sk = –Ck g p uslip

Suitable values for the model parameters Ck and C are not as well established as the parameters for single-phase flow. In the literature, values within the ranges

0.01 < Ck < 1 and 1 C 1.92 have been suggested (Ref. 1). The turbulent viscosity is taken into account in the momentum equations and by adding a drift velocity to the gas velocity:

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References for the Bubbly Flow Interfaces

1. A. Sokolichin, G. Eigenberger, and A. Lapin, “Simulations of Buoyancy Driven Bubbly Flow: Established Simplifications and Open Questions,” AIChE Journal, vol. 50, no. 1, pp. 24–49, 2004.

2.D. Kuzmin and S. Turek, Efficient Numerical Techniques for Flow Simulation in Bubble Column Reactors, Institute of Applied Mathematics, University of Dortmund, 2000.

3.C. Crowe, M. Sommerfeld, and Y. Tsuji, Multiphase Flows with Droplets and Particles, CRC Press, 1998.

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T h e o r y f o r t h e M i x t u r e M o d e l I n t e r f a c e

The mixture model is a macroscopic two-phase flow model, in many ways similar to the Bubbly Flow model. It tracks the averaged phase concentration, or volume fraction, and solves for one velocity field for each phase. It is suitable for mixtures consisting of solid particles or liquid droplets immersed in a liquid.

In this section:

•The Mixture Model Equations

•Dispersed Phase Boundary Conditions Equations

•Turbulence Modeling in Mixture Models

•Slip Velocity Models

•References for the Mixture Model Interfaces

The Mixture Model Equations

Just as the Bubbly Flow interface, The Mixture Model, Laminar Flow Interface is based on the two fluid Euler-Euler model. 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:

•The density of each phase is approximately constant.

•Both phases share the same pressure field.

•The relative velocity between the two phases is essentially determined assuming a balance between pressure, gravity, and viscous drag.

The momentum equation for the mixture is

ut + u u = – p – cd 1 – cd uslipuslip + Gm + g + F

where u denotes velocity (SI unit: m/s), density (SI unit: kg/m3), p pressure

(SI unit: Pa), cd mass fraction of the dispersed phase (SI unit: kg/kg), uslip the relative velocity between the two phases (SI unit: m/s), Gm the sum of viscous and turbulent

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stress (SI unit: kg/(m·s2)), g the gravity vector (SI unit: m/s2), and F additional volume forces (SI unit: N/m3). The velocity u used here is the mixture velocity (SI unit: m/s), defined as

c cuc + d dud

u = -------------------------------------------

where c and d denote the volume fractions of the continuous phase and the dispersed phase (SI unit: m3/m3), respectively, uc the continuous phase velocity (SI unit: m/s), ud the dispersed phase velocity (SI unit: m/s), c the continuous phase density (SI unit: kg/m3), d the dispersed phase density (SI unit: kg/m3), and the mixture density (SI unit: kg/m3). The relation between the velocities of the two phases is defined by

Here, uslip denotes the relative velocity between the two phases (SI unit: m/s). For different available models for the slip velocity, see Slip Velocity Models. 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.

The mixture density is given by

= c c + d d

where c and d are the densities of each of the two phases (SI unit: kg/m3). The mass fraction of the dispersed phase cd is given by

c = ----d------d-

d

The sum of viscous and turbulent stress is

Gm = + T u + uT

where is the mixture viscosity (SI unit: Pa·s), T the turbulent viscosity (SI unit: Pa·s). If no turbulence model is used, T equals zero.

The transport equation for d, the dispersed phase volume fraction, is

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