Review of Stability Criteria Based on Fundamental Approach

Review of Stability Criteria Based on Fundamental Approach 

In the previous section, stability criteria were obtained for gas-hquid bubble columns, gas-solid fluidized beds, liquid-sohd fluidized beds, and three-phase fluidized beds. Before we begin the review of previous work, let us summarize the parameters that are important for the fluid mechanical description of multiphase systems. The first and foremost is the dispersion coefficient. During the derivation of equations of continuity and motion for multiphase turbulent dispersions, correlation terms such as esv appeared [Eqs. (3) and (10)]. These terms were modeled according to the Boussinesq hypothesis [Eq. (4)], and thus the dispersion coefficients for the sohd phase and hquid phase appear in the final forms of equation of continuity and motion [Eqs. (5), (6), (14), and (15)]. However, for the creeping flow regime, the dispersion term is obviously not important. 

The second important term is the virtual mass coefficient (Cv). When the dispersed phase accelerates (or decelerates) with respect to the continuous phase, the surrounding continuous phase has to be accelerated (or decelerated). For such a motion, additional force is needed, which is called added or virtual mass force. This force was given by the second term in Eq. (8). The constant Cy is called the virtual or added mass coefficient. It is difficult to estimate the value of Cv with the present status of knowledge. Therefore, many recommendations are available in the published literature. In an extreme case of potential flow, the value of Cy is 0.5. 

As regards the pressure, at a given location the pressure has been considered to be the same for all the phases. Further, when any area is occupied by different phases, the pressure has been assumed to be shared by the phases proportional to the fractional area occupied by the phases. This approach has been used by several investigators in the past (Kataoka, 1986 Kataoka and Serizawa, 1989 Elgobashi and Abou-Arab, 1983). 

As regards to the estimation of force due to buoyancy, there have been two schools of thought. One considers the force equal to VpPtg where p, is the continuous phase density and Vp is the particle volume (Clift et al., 1987 Clift, 1993 Joshi, 1983 Epstein, 1984). The other school considers the buoyancy force equal to VpPog where pd is the average density of dispersion (Foscolo and Gibilaro, 1984 Gibilaro etal, 1987 Astarita, 1993). 

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I. Review of Stability Criteria Based on Fundamental Approach... 

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