Interphase momentum transfer

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

In order to obtain the interphase momentum transfer term in regions with a gas volume fraction less than 0,8 the Ergun equation is adapted ... 

Interphase momentum transfer is the focus of this section. Macroscopic correlations are based on dynamic forces due to momentum flux that act across the fluid-solid interface, similar to terms of type 2, 3, and 4 in the equation of motion. Gravity enters into this discussion via the hydrostatic contribution to fluid pressure, because volumetric body forces are not operative across an interface. The outward-directed unit normal vector from the solid surface into the fluid is n. As discussed earlier, forces due to total momentum flux, transmitted in the —n direction from the flnid to the solid across the interface at r = / , are (i.e., see equation 8-20) ... 

The interphase momentum transfer term can be derived from correlation developed to model fluidization processes since the range of solids concentrations experienced in pneumatic transport systems is similar. This form has been employed by Patel and Cross [46] for modeling gas-solids fluidized beds. For solid concentrations greater than 0.2, the interphase friction coefficient, K, may be computed by using the Ergun [47] equation ... 

The interphase momentum transfer coefficient P is frequently modeled by a combination of the Ergun equation and the Wen and Yu correlation, but in this model, the improved drag relation by Beetstra et al. (2007), based... 

Knowing 0, the solid phase pressure and the solid phase bulk and shear viscosities can be calculated from formulae derived by Tun et al. [1984]. The granular temperature conductivity, k, has also been formulated by Tun et al. [1984]. y has been modeled in terms of 0 by Jenkins and Savage [1983]. For dense regimes, the interphase momentum transfer coefficient, / , can be calculated from the Ergun equation already encountered in Chapter 11 on fixed bed reactors [Gidaspow, 1994]. For dilute regimes, a correlation has been proposed by Wen and Yu [1966]. 

The model species, total mass, momentum, and energy continuity equations are similar to those presented in Section 13.7 on fluidized bed reactors. Constant values of the gas and liquid phase densities, viscosities, and diffusivities were assumed, as well as constant values of the interphase mass transfer coefficient and the reaction rate coefficient. The interphase momentum transfer was modelled in terms of the Eotvos number as in Clift et al. [1978]. The Reynolds-Averaged Navier-Stokes approach was taken and a standard Computational Fluid Dynamics solver was used. In the continuous liquid phase, turbulence, that is, fluctuations in the flow field at the micro-scale, was accounted for using a standard single phase k-e model (see Chapter 12). Its applicability has been considered in detail by Sokolichin and Eigenberger [1999]. No turbulence model was used for the dispersed gas phase. Meso-scale fluctuations around the statistically stationary state occur and were explicitly calculated. This requires a transient simulation and sufficiently fine spatial and temporal grids. 

In the drag force, the influence of the gas on the particle is described with interphase momentum transfer coefficient P p which is calculated depending on the porosity of the soHd phase e. For the dense regimes, when porosity is smaller than 0.8, the coefficient is calculated according to the Ergun correlation (Ergun, 1952) ... 

Because the interphase momentum transfer is usually dominant, the interphase forces can best be treated in a semi-imphcit manner. In particular, the last term on the RHS can be combined with the LHS to obtain ... 

---