Coefficients virtual mass coefficient

The added mass force accounts for the resistance of the fluid mass that is moving at the same acceleration as the particle. Neglecting the effect of the particle concentration on the virtual-mass coefficient, for a spherical particle, the volume of the added mass is equal to one-half of the particle volume, so that... 

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. 

Various model parameters involved in the derivation of the stability criterion need to be specified in order to use the stability criterion for quantitative predictions. Model parameters essential for this purpose include the slip velocity, the virtual mass coefficient, and the dispersion coefficient. The procedure for estimation of these parameters is given for gas-solid (and solid-liquid) fluidized beds and bubble columns. 

Jackson (1985) used a constant value of 0.5 as the virtual mass coefficient, which is for an isolated sphere in an infinite medium. Homsy et al. (1980)... 

Physical properties of the fluid such as density, viscosity, and particle density and the model parameters such as dispersion coefficient and virtual mass coefficient have a substantial effect on the critical diameter. These effects are discussed systematically in the following paragraphs. 

In the case of gas-solid fluidized beds, the effect of virtual mass coefficient is negligible as shown in Fig. 22. However, as the gas density increases the effect of virtual mass becomes similar to that observed in solid-liquid systems as shown in Fig. 23. 

Fig. 21. Effect of formulation of virtual mass coefficient on lower particle critical diameter solid-liquid fluidized beds a = 3.0, /tl = 1 mPas, pi = 1000 kg/m ].

Equation (25) was used to obtain the critical transition gas hold-up. Critical gas transition hold-up is plotted against terminal bubble rise velocity in a typical stability map. The effects of various parameters such as virtual mass coefficient, Richardson-Zaki index, and proportionality constant for dispersion are described next. 

Fig. 25. Effect of virtual mass coefficient on transition gas hold-up bubble columns [a = 0.5, m = 1.9, dB-VBoo Clift et al relation].

The material derivative, D/Dt, in this equation should be the derivative pertaining to the dispersed phase particle. The virtual mass coefficient, Cvm, may be a function of the volume fraction of neighboring bubbles. For a single dispersed particle, it is in the range 0.25 to 0.5. For gas-liquid flows, van Wijngaarden (1976) recommended following expression to estimate Cym ... 

The virtual mass coefficient for a sphere in an invicid fluid is thus Cy = The basic model (5.111) is often slightly extended to take into account the self-motion of the fluid. In general the added mass force is expressed in terms of the relative acceleration of the fluid with respect to the particle acceleration. 

The virtual mass coefficient Cgp of an isolated spherical bubble is 0.5. The BP gradient is then added to the right-hand side of the gas momentum (Eq. (5.4)) and acts as a driving force for bubbles to move from areas of higher to areas of lower d and facilitates stabilization of the bubbly flow regime. However, Sankaranarayanan and Sundaresan (2002) indicate that as increases, the collisional and hydrodynamic contributions become important. 

Cl, CvM drag, lift, and virtual mass coefficients (—), respectively... 

The virtual mass coefficient for a sphere in an invicid fiuid is thus Cv =. ... 

In the case of solid-liquid fluidized beds, the effect of virtual mass is to make the bed more unstable as shown in Fig. 21. This can be explained as follows The effect of virtual mass is to increase the apparent density of the particle. As discussed in this section earlier, an increase in the particle density makes the system more unstable. This observation is consistent with Fig. 21. As the particle density increases, say, ps - 9000 kg/m and Pl = 1000 kg/m, the effect of virtual mass is negligible and the curves are seen to merge irrespective of the formulation of virtual mass coefficient. 

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