Virtual-mass force effective volume

It is very interesting to note that the virtual-mass force mainly acts through the effective volume coefficient y, and tends to reduce the momenffim-exchange terms. The final step is to use the definitions of the mass-average moments introduced in Chapter 4 to derive the force terms in the macroscale momenffim balance. For the disperse-phase momentum balance (see Eq. (4.85) on page 123), this procedure leads to... 

As can be seen from Eq. (5.100), the virtual-mass force reduces the drag and lift forces by a factor of 1 /y. The buoyancy force is not modified because we have chosen to define it in terms of the effective volume Vpy. We remind the reader that the mesoscale acceleration model for the fluid seen by the particle A j must be consistent with the mesoscale model for the particle phase A p in order to ensure that the overall system conserves momentum at the mesoscale. (See Section 4.3.8 for more details.) As discussed near Eq. (5.14) on page 144, this is accomplished in the single-particle model by constraining the model for Apf given the model for Afp (which is derived from the force terms introduced in this section). Thus, as in Eqs. (5.98) and (5.99), it is not necessary to derive separate models for the momentum-transfer terms appearing in Apf. 

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... 

At low gas volume fraction (<0.01), the forces acting on bubbles in a liquid are similar to those acting on a single bubble. Due to their low inertia as compared with the Hquid phase, bubbles are subject to a large number of forces, i.e., forces due to drag, lift, and virtual mass. In the definition of the interfacial forces, it is customary to characterize the bubble size with the sphere equivalent diameter. That is, ah effects due to nonsphericity are lumped in the closure. The closures that are most commonly applied... 

In particular, referring to the introduction of the external forces as presented in sect 4.1.3 there are still no complete consensus in the literature regarding the treatment of the interfacial coupling terms like the steady drag-, added mass- and lift forces. In one view it is considered convenient to split the net force exerted by the interstitial fluid on the particle into two different contributions One virtual force applied by an undisturbed flow on a imaginary fluid particle which coincides with the solid particle in volume and shape, and a second contribution that represents the forces due to the perturbations in the flow. These flow disturbances are created by the presence of the particles. The phrase undisturbed flow thus refers to the flow that would be observed if the particle was not present. Neglecting the effects of the perturbations in the flow, the net force exerted on a particle (4.57) might be approximated by ... 

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