Assessment of filtered gas–solid momentum transfer models via a linear wave propagation speed test

Abstract

The propagation speeds of linear waves in gas–solid suspensions depend strongly on the solids volume fraction and the wave frequency. The latter is due to gas–solid momentum transfer and allows a simple test on filtered gas–solid momentum transfer models. Such models may predict linear wave propagation speeds different from those obtained with the non-filtered model at wave frequencies higher than the filter frequency, but not at wave frequencies lower than the filter frequency. For the filtered drag, an effective drag coefficient approach is shown to alter the linear wave propagation speeds in the entire wave frequency range, independent of the applied effective drag coefficient. Furthermore, as the effective drag coefficient decreases, the high frequency linear wave propagation speeds are gradually introduced at lower wave frequencies. For the filtered momentum transfer due to the correlation between the solids volume fraction and the gas phase pressure gradient, the behavior of an apparent added mass closure model and an apparent history force closure model are investigated. An apparent added mass introduces the filter frequency linear wave propagation speeds to frequencies higher than the filter frequency. The linear wave propagation speeds for wave frequencies lower than the filter frequency are, however, not altered. Furthermore, an apparent added mass introduces no intrinsic wave frequency dependence in the linear wave propagation speeds, in agreement with its source term in the non-filtered model. Hence, the frequency dependence of the linear wave propagation speeds at frequencies lower than the filter frequency is still to be provided by a drag type term. As such, the behavior introduced by an apparent added mass is acceptable for filtered models. Also, to a certain extent, an apparent added mass can restore the linear wave propagation speed behavior at wave frequencies lower than the filter frequency altered by an effective drag coefficient approach. The reformulation of the apparent added mass in terms of an apparent distribution of the filtered gas phase pressure gradient over the phases and an apparent (effective) drag force is investigated. An apparent history force introduces intrinsic wave frequency dependence in the linear wave propagation speeds and alters the latter from the low wave frequencies on. As such, the behavior introduced by an apparent history force is unacceptable for filtered models.