(137b) The Effective Drag Closure for Gas-Solid Flows in Vertical Risers | AIChE

(137b) The Effective Drag Closure for Gas-Solid Flows in Vertical Risers


Özel, A. - Presenter, Institut de Mécanique des Fluides de Toulouse
Fede, P. - Presenter, Université Paul Sabatier
Simonin, O. - Presenter, Institut de Mécanique des Fluides de Toulouse

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The Eulerian multi-fluid approach is a powerful numerical tool to investigate high-velocity gas-solid flows in vertical risers. The overall hydrodynamic behaviors of gas-solid flows are significantly modified by the existence of mesoscales, such as streamers and clusters in dilute flows. Clusters and streamers continuously occur in risers by a reason of local instabilities, such as damping of fluctuating motion of particles by the interstitial fluid, inelastic collisions, the non-linear drag between phases (Agrawal et al., 2001). The typical length scale of mesoscales is in the order of $10-50$ particle diameters and the time scale varies between $0.001~s$ and $0.1~s$ (De Wilde, 2005). Although formations and breakage of mesoscales structures can be captured by multi-fluid formalism and transport equations developed in the frame of the kinetic theory of granular media on a small domain, large ranges of spatial and temporal variations of these structures restrict the resolution of calculations for large industrial units due to computational cost. To account for the effect of mesoscales on the macroscopic behavior in coarse grid simulations, the analogy can be constructed with large eddy simulation of single phase turbulent flow. These effects can be taken into account by subgrid scales through additional closure relations. The key feature in the present study is the proposition of the subgrid model for monodisperse gas-solid flow which takes into account the effects of mesoscales and validation of this model by experimental data.

In this paper, the modelling approach is based on the multi-fluid model formalism that involves mean separate transport equations of mass, momentum and energy for each phases. Interactions between phases are coupled through interphase transfers. Two transport equations, developed in the frame of kinetic theory of granular media supplemented by the interstitial fluid (Balzer et al., 1996), were solved to model the effects of velocity fluctuations and inter-particle collisions on the dispersed phase hydrodynamic. The gas phase is predicted using Large Eddy Simulation. Concerning the transfers between the phases with non-reactive isothermal flow, the drag and Archimede forces are accounting for the momentum transfer. Three-dimensional simulations are performed by the numerical code - $NEPTUNE\_CFD\,\,v1.07@Tlse$ - of a french industrial consortium comprised of EDF / CEA / AREVA_NP / IRSN, which has been extended by Institut de Mécanique des Fluides de Toulouse (IMFT). The point has to be stated that this study does not discuss the multi-fluid model, only proposes a different closure equation for an effective drag coefficient for the interfacial momentum transfer which is explicitly filtered by the grid size.

The effects of the mesoscale structures have been investigated by several authors. The pioneer work done by Agrawal et al. (2001) and it is remarked that coarse grid simulations overestimate the drag force. Wang (2006) proposed a structured dependent drag force by the energy-minimization multi-scale model. Addition to these works, Heynderick (2004) has taken into account the clusters effect by the concept of effective drag with considering the solid particles to be a part of a cluster and also to be present as individual particles. Igci (2007) proposes filter size dependent closures for the effective drag.

In the present modelling approach, we propose the following decomposition of the filtered interfacial momentum transfer:

\begin{displaymath}  \overline{\bf {I}}_g=-\overline{\bf {I}}_p=\overline{\frac{\...  ...\left(\tilde{\bf {U}}_p-\tilde{\bf {U}}_g+{\bf {V}}_d^*\right)  \end{displaymath} (1)

where $\overline{\alpha_p}$ is the filtered solid volume fraction (the "overline" defines the filtered quantity),

\begin{displaymath}  \tilde{U}_p=\frac{\overline{\alpha_p~U_p}}{\overline{\alpha...  ...tilde{U}_g=\frac{\overline{\alpha_g~U_g}}{\overline{\alpha}_g}  \end{displaymath} (2)

In ([*]), $\tilde{\tau}_{gp}^F$ is the particle relaxation time scale. The velocity $\bf {V}_d^*$ is the subgrid contribution that we propose to model as:

\begin{displaymath}  \bf {V}_d^*=\ln{\left[\frac{{\Delta}^*}{1+{\Delta}^*}\right]...  ...ta}^*}}\times  \left(\tilde{\bf {U}}_p-\tilde{\bf {U}}_g\right)  \end{displaymath} (3)

where the non-dimensional grid size, ${\Delta}^*$, is defined as follows:

\begin{displaymath}  {\Delta}^*= \overline{\alpha_p}~\frac{\Delta}{{\left({\frac{...  ...t\tilde{\bf {U}}_p-\tilde{\bf {U}}_g\right\vert}\right)}^{3}}  \end{displaymath} (4)

where $\Delta$ is the control volume of cell. The subgrid model construction has been done by "a priori" tests. High-resolution numerical predictions have been computed with fine computational grid ensuring results without dependency on the grid resolution. These high-resolution data are filtered by several filter widths based on explicit grid sizes. Then, the subgrid drift velocity has been constructed by difference between filtered and high-resolution data.

The results obtained by the subgrid model are validated with experimental data of Andreux (2008) and Andreux (2001). The riser is a scaled cold circulating fluidized bed of $\unit{10~m}$ high having a square cross section of $\unit{11~cm~x~11~cm}$. Typical FCC particles (A-type) with density $\unit{1400~kg/m^3}$ and mean diameter $\unit{70~\mu m}$ were conducted. The density of air is set to $\unit{1.2~kg/m^3}$ and dynamic viscosity is equal to $\unit{1.8 \cdot 10^{-5} Pa \cdot s}$. Averaged volume fractions of solid and pressure gradients obtained with and without subgrid model are compared in Fig.[*]. The unphysical accumulation at the bottom of the riser vanishes and the pressure gradient obtained by fine mesh is in a good agreement with the one obtained by subgrid model.

Figure: (a) Averaged volume fraction solid distribution and (b) pressure gradient along streamwise direction.

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OZEL Ali 2009-05-11


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