Virtual mass effect

The virtual mass effect relates to the force required for a particle to accelerate the surrounding fluid [65, 170, 26]. When a particle is accelerated through a fluid, the surrounding fluid in the immediate vicinity of the particle will also be accelerated at the expense of work done by the particle. The particle apparently behaves as if it has a larger mass than the actual mass, thus the net force acting on the particle due to this effect has been called virtual mass or added mass force. The steady drag force model does not include these transient effects. 

Drew, D. A., Cheng, L. Lahey, R. T. 1979 The analysis of virtual mass effects in two-phase flow. International Journal of Multiphase Flow 5, 233-242. 

When fluids can seep through pores, interacting mechanically with the solid skeleton, the material is composed of more than one constituent thus we need to use a mixture theory in which we could clearly make out each part filled by different constituents on a scale which is rather large in comparison with molecular dimensions so we put forward a new continuum theory of an immiscible mixture consisting both of a continuum with ellipsoidal microstructure (the porous elastic solid) and of two classical media (see, also, the conservative case examined by Giovine (2000)). In accordance with Biot (1956), we consider virtual mass effects due to diffusion we also introduce the microinertia associated with the rates of change of the constituents local densities, as well as the one due to the deformation of the pores close to their boundaries. 

Other forces can be considered in order to take into account the lift resulting from the velocity gradient around the droplet or the virtual mass effect. However, in SCR simulations they can be neglected, since their contribution is usually from 2 to 3 order of magnimdes lower than the others [39]. A typical expression for the determination of the drag coefficient Cd is ... 

We have so far described drag and lift forces acting on a suspended particle. There are, however, additional hydrodynamic forces, such as Basset history, Faxen correction, and virtual mass effects that act on the particles. Some of these forces could become important especially for the particles suspended in a liquid. The general equation of motion of a small spherical particle suspended in fluid as obtained by Maxey and Riley is given as... 

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... 

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. 

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 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. 

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 virtual mass terms are not considered. In bounded case, the disturbances (destabilizing effects) due to the sparger are present. 

This added mass effect is modeled by introducing the virtual mass term, Fvm ... 

Oil residuals are the dominating sensory pollution source in new ducts. The sensory assessments showed a clear correlation between the total mass of oil residuals (average surface density times surface area) and the PAP (Fig. 7). Emissions from dust/debris accumulated in the ducts during construction (mostly inorganic substances) seem to be less important. No simple correlation was observed between the amount of accumulated dust and odour emissions however, the organic dust accumulated during the operation period may produce more severe odour emissions. When dust had accumulated on the inner surface of the ducts, the relative humidity of the air in the ducts had an effect. On the other hand, the relative humidity had virtually no effect on the odour emissions of oil residuals. 

It is emphasized that the virtual mass force accounts for the form drag (shape effects) due to the relative acceleration between the particle and the surrounding fluid. 

While the virtual mass force accounts for the form drag on the particle due to relative acceleration between the particle and the surrounding fluid, the history term accounts for the corresponding viscous effects. Moreover, the history force originates from the unsteady diffusion of the vorticity around the particle so there is a delay in the boundary layer development as the relative velocity changes with time [96, 97, 22]. This means that when the relative velocity between the particle and the fluid varies, the vorticity present at the particle surface changes and the surrounding flow needs a flnite time to readapt to the new conditions. 

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