Virtual mass

Bubbly flow Instead of assuming that dS/dp = 0 as in his earlier model, Henry proposed that the virtual mass term of the gaseous volumes should be included ... 

Annular flow, wavy interface (Henry, 1971) Under this condition the virtual mass of gas flowing over a wavy surface can be approximated by flow over a surface made of continuous rows of half-cylinders ... 

Many engineering operations involve the separation of solid particles from fluids, in which the motion of the particles is a result of a gravitational (or other potential) force. To illustrate this, consider a spherical solid particle with diameter d and density ps, surrounded by a fluid of density p and viscosity /z, which is released and begins to fall (in the x = — z direction) under the influence of gravity. A momentum balance on the particle is simply T,FX = max, where the forces include gravity acting on the solid (T g), the buoyant force due to the fluid (Fb), and the drag exerted by the fluid (FD). The inertial term involves the product of the acceleration (ax = dVx/dt) and the mass (m). The mass that is accelerated includes that of the solid (ms) as well as the virtual mass (m() of the fluid that is displaced by the body as it accelerates. It can be shown that the latter is equal to one-half of the total mass of the displaced fluid, i.e., mf = jms(p/ps). Thus the momentum balance becomes... 

The fluid-particle interaction force, omitting the virtual mass term and combining the pressure terms in the equation of motion becomes... 

Other than the virtual momentum force Fp, a virtual mass source/sink should also be applied to the particle surface to satisfy the continuity for the control volume containing the particle surface or the particle (Kim et al., 2001). The mass source can be calculated by... 

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... 

Note that for liquid solid systems, Eq. (20) should also include the short-range lubrication forces and the effects of other forces such as the virtual mass force. But this is beyond the scope of this chapter. 

One therefore has to decide here which components of the phase interaction force (drag, virtual mass, Saffman lift, Magnus, history, stress gradients) are relevant and should be incorporated in the two sets of NS equations. The reader is referred to more specific literature, such as Oey et al. (2003), for reports on the effects of ignoring certain components of the interaction force in the two-fluid approach. The question how to model in the two-fluid formulation (lateral) dispersion of bubbles, drops, and particles in swarms is relevant... 

There has been no systematic study of the influence of gas properties. The gas density appears in the analysis either in the virtual mass term or in... 

Davidson and Harrison (D6) have brought about a change in Eq. (1) by using a value of 1/2 in the place of 11/16 for the virtual mass term. Proceeding in exactly the same manner as earlier, they obtained... 

The other force resulting from expansion can be computed from the rate of change of momentum of the forming bubble, i.e., (djdte)(Mve). Here M is the virtual mass of the bubble and is the sum of the mass of the gas in the bubble and that of 11/16 ths of its volume of the liquid surrounding it, and ve is the velocity of expansion of the bubble. Thus,... 

In Eq. (103) the value of 1/2 has been used for evaluating the virtual mass of the bubble. 

Here, the virtual mass M of the drop can be assumed to be constant and equal to that of the static drop, because there is not much growth before detachment. The mass term is written as... 

Virtual mass = Mass of particle + Added mass. 

As we have conservation of velocity, i.e., vi = V2 = v, the momentum of a fragment ion mf formed in a FFR preceding the magnetic sector is different from that of such a fragment ion arising from the ion source. The ion formed by metastable ion dissociation thus passes the magnet as if it had the virtual mass m ... 

The first term of Eq. (11-11) is the Stokes drag for steady motion at the instantaneous velocity. The second term is the added mass or virtual mass contribution which arises because acceleration of the particle requires acceleration of the fluid. The volume of the added mass of fluid is 0.5 F, the same as obtained from potential flow theory. In general, the instantaneous drag depends not only on the instantaneous velocities and accelerations, but also on conditions which prevailed during development of the flow. The final term in Eq. (11-11) includes the Basset history integral, in which past acceleration is included, weighted as t — 5) , where (t — s) is the time elapsed since the past acceleration. The form of the history integral results from diffusion of vorticity from the particle. 

Trapping of liquid in the rough surface of the electrode adds virtual mass and may also cause an additional mass loading artifact (Theisen et al., 2004). This effect can be particularly severe when porous materials such as conducting polymers are deposited at the QCM electrode. 

The acting forces per unit volume between the phases are due to drag and virtual-mass. 

In the preceding equations, Fp can be expressed as a combination of local averaged drag force and virtual mass force [Anderson and Jackson, 1967]. 

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. 

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