(140h) The Impact of Frictional Stresses on the Sub-Grid Modification for Gas–Solid Drag

Fluidized beds are widely used in a variety of industrially important processes. During the last decades the analysis of the hydrodynamics or the efficiency of fluidized beds through numerical simulations has become increasingly common.¹ During the last decades the analysis of the hydrodynamics or the efficiency of fluidized beds through numerical simulations has become increasingly common, where the two-fluid model (TFM) approach has proven to provide fairly good predictions of the hydrodynamics of gas-solid flows.²

However, due to computational limitations a fully resolved simulation of industrial scale reactors is still unfeasible. It is, therefore, common to use coarse grids to reduce the demand on computational resources, which inevitably neglects small (unresolved) scales.³ Many sub-grid drag modifications have, therefore, been put forth by academic researchers to account for the effect of small unresolved scales on the resolved meso-scales in this case.^1,4,5

Our previous studies^1,4 reveal that the state-of-the-art sub-grid drag modifications^5–8 show different functional dependencies as well as completely different functional forms. For example, while EMMS⁷ and the Kuipers⁶ relation do not show a grid dependency, the other drag modifications predict a reduction of the effective drag with increasing grid/filter size. Furthermore, the filtered relation of Igci et al.⁹ does not reveal a functional dependence on the filtered slip velocity (in a more recent publication the group of Sundaresan added one⁵). Surprisingly the EMMS model reveals a correction at maximum packing.¹ Finally, even drag modifications^4,5,8 derived from filtering fine grid simulations reveal significantly different forms, while the functional dependencies and trends seem to be quite similar. Hence, the question arises, where these differences come from?

First, it has to be emphasized that these drag modifications are based on different philosophies. The EMMS,⁷ the CD-Lab¹ and the Kuipers⁶ drag modifications are derived from physical assumptions, that is on cluster/bubble formation. In contrast, the modifications proposed by Milioli et al.,⁵ Parmentier et al.⁸ and Schneiderbauer et al.⁴ are derived from filtering fine grid simulations. Second, the drag corrections are usually dedicated to different particle sizes (size < 100mm,^5,7,8,10; size >> 100mm,^1,4,6) and flow regimes.

These filtered models are commonly deduced from different simulation setups and constitutive relations used for the fine grid simulations. In this paper, we concentrate on the impact of the frictional stress modeling on the functional forms of the sub-grid modifications. In particular, we show that close to the maximum packing the frictional stresses tend to homogenize the granular system and, therefore, less reduction of the drag is observed in that area, which is also indicated by a reduction of the amount of meso-scale velocity fluctuations with increasing frictional stresses. However, this effect is relatively small and appears to be sensitive to the filtered slip velocity and the filter lengths. In particular, at large filter length and large slip velocities the impact of the frictional stresses vanishes. For these high slip velocities the frictional stresses only have a minor impact on the local inhomogeneity of the granular system, since these stresses are orders of magnitude smaller than the gas-solid drag.

References:

1.        Schneiderbauer, S., Puttinger, S. & Pirker, S. Comparative Analysis of Subgrid Drag Modifications for Dense Gas-Particle Flows in Bubbling Fluidized Beds. AIChE J. 59, 4077–4099 (2013).

2.        Schneiderbauer, S., Aigner, A. & Pirker, S. A comprehensive frictional-kinetic model for gas-particle flows: analysis of fluidized and moving bed regimes. Chem. Eng. Sci. 80, 279–292 (2012).

3.        Agrawal, K., Loezos, P. N., Syamlal, M. & Sundaresan, S. The role of meso-scale structures in rapid gas-solid flows. J. Fluid Mech. 445, 151–185 (2001).

4.        Schneiderbauer, S. & Pirker, S. Filtered and heterogeneity based sub-grid modifications for gas-solid drag and solids stresses in bubbling fluidized beds. AIChE J. 60, 839–854 (2014).

5.        Milioli, C. C., Milioli, F. E., Holloway, W., Agrawal, K. & Sundaresan, S. Filtered two-fluid models of fluidized gas-particle flows: new constitutive relations. AIChE J. 59, 3265–3275 (2013).

6.        Wang, J., van der Hoef, M. A. & Kuipers, J. A. M. Coarse grid simulation of bed expansion characteristics of industrial-scale gas-solid bubbling fluidized beds. Chem. Eng. Sci. 65, 2125–2131 (2010).

7.        Lu, B., Wang, W. & Li, J. Searching for a mesh-independent sub-grid model for CFD simulation of gas-solid riser flows. Chem. Eng. Sci. 64, 3437–3447 (2009).

8.        Parmentier, J.-F., Simonin, O. & Delsart, O. A functional subgrid drift velocity model for filtered drag prediction in dense fluidized bed. AIChE J. 58, 1084–1098 (2012).

9.        Igci, Y. & Sundaresan, S. Constitutive Models for Filtered Two-Fluid Models of Fluidized Gas-Particle Flows. Ind. Eng. Chem. Res. 50, 13190–13201 (2011).

10.      Igci, Y. & Sundaresan, S. Verification of filtered two-fluid models for gas-particle flows in risers. AIChE J. 57, 2691–2707 (2011).