Reformulating and quantifying the generalized added mass in filtered gas-solid flow models

Abstract

To account for mesoscale phenomena in coarse grid simulations, Reynolds stress terms appearing in the filtered gas-solid flow equations have to be modeled. A generalized added mass approach previously proposed by Zhang and VanderHeyden ͓Int. J. Multiphase Flow 28, 805 ͑2002͔͒ to model the acoustic gas-solid interaction Reynolds stress term is analyzed. Theoretically, it is shown that a generalized added mass term appears directly from the filtered acoustic gas-solid interaction term and that it corresponds to a redistribution of the filtered gas phase pressure gradient over the phases. This direct contribution scales according to the mean square of the solid volume fraction fluctuations. Two-dimensional dynamic mesoscale simulations over a broad solid volume fraction range and for two domain sizes and two grid resolutions are carried out to calculate the magnitude of the generalized added mass effect. Calculated values of the mean square of the solid volume fraction fluctuations are qualitatively in agreement with the experimental observations of Zenit and Hunt ͓Int. J. Multiphase Flow 26, 763 ͑2000͔͒. A second, indirect contribution to the generalized added mass term from the filtered acoustic gas-solid interaction term is shown to be statistically significant, but one order of magnitude smaller than the direct contribution. A further quantification of the maximum generalized added mass effect as a function of the filter frequency is obtained from a mixture speed of sound test. The results show that a large generalized added mass coefficient, as previously reported by Zhang and VanderHeyden ͓Int. J. Multiphase Flow 28, 805 ͑2002͔͒, is justified only in case the filter frequency is low ͑Ͻ20 Hz͒, i.e., if the grid is, either spatially or temporally, sufficiently coarse.

Length scale dependence of effective inter-phase slip velocity and heterogeneity in gas–solid suspensions

“…It can be seen that after an induction period with uniform states, the system is soon destabilized by the formation of clustering structure of particles on meso-scales as shown in Fig. 2, which results in the fluctuation of U slip and H d around their mean value much larger than the corresponding values in the uniform states, the same tendency is reported previously (Agrawal et al, 2001;Andrews IV et al, 2005;De Wilde, 2005). Here, the simulated domains were comparable in size to the computational cells used in coarse grid simulations of industrial scale equipments.…”

“…For molecular fluids not subjected to large shear rates or large thermal gradients, there is a plateau of scales between the mean free path of the molecules and the minimally characteristic scales of macroscopic properties, where scaleindependent mass, momentum and energy densities and fluxes can be defined (Goldhirsch, 1999). However, many recent studies have shown that such plateau is non-existent in gas-solid two-phase flow, and the lack of scale-separation in both space and time were found (Agrawal et al, 2001;Andrews IV et al, 2005;De Wilde, 2005;Sundaresan, 2000;van der Hoef et al, 2006). These studies have shown that the effective inter-phase drag coefficient (or effective inter-phase slip velocity) and other properties are significantly dependent on the size of the periodic domain used in computer simulations.…”

Filtered two‐fluid models for fluidized gas‐particle suspensions

“…Zhang and VanderHeyden 42 and de Wilde 43,44 argue that À/ 0 s rp 0 g should also include a dynamic part (namely, an added mass force). Andrews 45 found in his simulation study that the principal contribution of À/ 0 s rp 0 g was to a filtered drag force term, which has been included in our study.…”

“…The / 0 s r Á r 0 g term in Eq. 20 can also add a dynamic part, resembling an apparent added mass force [42][43][44] ; however, as Andrews 45 found such a dynamic part to be much smaller than the drag force term in Eq. 20, we will limit ourselves to Eq.…”

Rotating fluidized bed with a static geometry: Guidelines for design and operating conditions

“…1) (Anderson et al, 1972;Volchkov et al, 1993;Goldshtik et al, 1997;de Broqueville, 2005de Broqueville, , 2006aDe Wilde et al, 2006, 2007aDe Wilde andde Broqueville, 2007, 2008). The fluidization gas is forced to leave the fluidization chamber via a central chimney, introducing a radial motion of the gas.…”