Added-mass force

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... 

In particular applications alternative relations for the slip velocity (3.428) can be derived introducing suitable simplifying assumptions about the dispersed phase momentum equations comparing the relative importance of the pressure gradient, the drag force, the added mass force, the Basset force, the Magnus force and the Saffman lift force [125, 119, 58]. For gas-liquid flows it is frequently assumed that the last four effects are negligible [201, 19[. 

Ramshaw and Trapp [172] incorporated surface tension effects, which had traditionally been ignored. Travis et al ]220] considered viscous stresses, La-hey et al [134] considered the added mass force, Prosperetti and Wijngaar-den [167] considered compressibility effects. Trapp [219] considered Reynolds stresses, while Stuhmiller [213], Boure ]22], Pauchon and Banerjee ]161, 162], Sha and Soo [188], Prosperetti and Jones ]168] and Holm and Kupershmidt [109] have put their main focus on the adoption and interpretation of the interfacial pressure forces (see sects 3.4.1 and 3.4.6). The simplest choice for the interfacial pressure distribution pk)A, is to assume that it is equal to the fluid bulk pressure. This implies that (pg)vg = pi)vt = Pq)ai = pi)ai for a two-phase system and as a result the current engineering practice is obtained (e.g., ]168, 125, 119]). However, this approach leads to an equation set involving a single pressure, which has real characteristics only when the two fluid velocities are equal [219]. 

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. 

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. 

Alternative, and supposedly more advanced, added mass force and coefficient formulations have been presented and discussed in the literature [31, 32, 24, 164, 170, 19, 51, 94]. However, there are still controversis regarding the physics of added mass in particular considering dense dispersions [158]. 

Implementing the added mass force has barely any influence on the steady state solution [30, 66]. Been et al [30] explained this to some extent surprising result by the fact that the simulations soon reach a quasi-stationary state where there is only minor acceleration. The bubble jets observed close to the distributor plate are then disregarded. However, the convergence rate and thus the computational costs are often significantly improved implementing this force. 

Deen et al [30] used the lift force in addition to the steady drag- and added mass forces in their dynamic 3D-model to obtain the transversal spreading of the bubble plume which is observed in experiments. A prescribed zero void wall boundary was used forcing the gas to migrate away from the wall. The continuous phase turbulence was incorporated in two different ways, using... 

Fig. 8.9. Axial velocity-, gas voidage- and turbulent viscosity profiles as a function of column radius at the axial level z = 2.0 (m) after 80 (s) (steady-state) employing the steady drag and added mass forces. Crosses experimental data [61], continuous line standard k-s model, case (a), dotted hne standard k-s model plus Sato model, case (b), dashed line extended k-s model, case (c). Grid resolution 20x72, time resolution 2 10 " (s). Reprinted with permission from [66]. Copyright 2005 American Chemical Society.

The acceleration of the liquid in the wake of the bubbles can be taken into account through the added mass force given by (5.112), whereas the Eulerian lift force acting on the dispersed phase is normally expressed on the form (5.65). 

The interphase forces considered were steady drag, added (virtual) mass and lift. The steady drag force on a collection of dispersed bubbles with a given average diameter was described by (5.48) and (5.34). The transversal lift force was determined by the conventional model (5.65), whereas the added mass force was approximated by (5.112). 

There is no gravity force in radial direction. Added Mass force ... 

The added mass force term is treated in the same way as described for the radial liquid velocity component. 

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