A comprehensive frictional-kinetic model for gas–particle flows: Analysis of fluidized and moving bed regimes
Highlights
► The paper describes a comprehensive frictional-kinetic multi-fluid model. ► A is used to close the frictional shear stresses. ► The model also accounts for dilation during frictional contacts under shear. ► The model shows excellent agreement with measurements in both regimes. ► The impact of the form of the solids wall shear stresses on the results is shown.
Introduction
Fluidized beds and moving beds are widely used in process industries, for example, for biomass reactors (Papadikis et al., 2008, Papadikis et al., 2009, Xue et al., 2011), polymerization reactors (Rokkam et al., 2010), metallurgical processes (Schatz, 2000) and for the discharge of granular materials from silos (Beverloo et al., 1961, Nedderman et al., 1982, Rycroft, 2007, Benyahia, 2008). The favorable properties of fluidized beds, that is a high degree of particle mixing and the effective particle–fluid heat transfer, have been known even since the forties of the twentieth century (Gilliland and Mason, 1949, Mickley and Trilling, 1949, Mickley and Fairbanks, 1955). The investigation of these was, however, restricted to simple experimental techniques and analytical considerations. In the past decades increasing computer power has become available and the numerical simulation of fluidized beds has become feasible opening a third branch to investigate bubbling fluidized beds (Syamlal et al., 1993, Enwald et al., 1996, van Wachem et al., 1999, van Wachem et al., 2001, Almuttahar and Taghipour, 2008, Reuge et al., 2008, Li et al., 2010, Passalacqua and Fox, 2011), spout beds (Du et al., 2006; Gryczka et al., 2009a, Gryczka et al., 2009b; Chen et al., 2011) and moving beds (Srivastava and Sundaresan, 2003, Benyahia, 2008).
In these systems the behavior of the interstitial gas, the gas solid interaction (i.e. drag) and the particle–particle contacts (binary and frictional) are the physically most relevant effects. One of the most appropriate numerical models for the simulation of gas–solid flows is CFD-DEM (Link et al., 2005, Zhu et al., 2007, van der Hoef et al., 2008, Godlieb, 2010, Laverman, 2010, van Buijtenen et al., 2011, Goniva et al., 2011), where the interstitial gas is modeled by a continuum approach, e.g. computational fluid dynamics (CFD), and the granular material by a discrete element method (DEM). In DEM each particle trajectory as well as each particle–particle collision is resolved. Recently, huge effort is made in developing parcel-based DEM methods (O'Rouke and Snider, 2010, Radl et al., 2011 and references therein) reducing the computational effort of the grain based part significantly.
Since the total number of particles involved in most practically relevant particulate flows is extremely large, it may be impractical to solve the equations of motion for each particle (Agrawal et al., 2001). It is, therefore, common to investigate particulate flows in large process units using averaged equations of motion (Anderson and Jackson, 1967, Ishii, 1975). For solids volume fractions up to 40% (), which is a typical range in fluidized beds and risers, the solids stresses arising from particle–particle collisions and the translational dispersion of the grains are commonly deduced by adapting the kinetic theory of gases (Chapman and Cowling, 1970, Lun et al., 1984, Rao and Nott, 2008). The derivation of this kinetic theory of granular flows (KTGF), thereby, assumes that the collisions can be considered as binary and that multiple collisions are rare. However, in quasi-static areas of fluidized beds and especially in moving beds the volume fraction exceeds 40%. In this regime, which is referred to as frictional (, Forterre and Pouliquen, 2008), the particle–particle contacts are determined largely by enduring multiple frictional contacts (Zhang and Rauenzahn, 1997) and, thus, kinetic theory does not apply.
The closure of these dominant sustained multiple sliding frictional contacts in the dense regime is commonly based on critical state theory of soil mechanics (Srivastava and Sundaresan, 2003 and references cited therein). In the literature there is a huge amount of studies dealing with frictional stress closures, which are based on Coulomb's law (Schaeffer, 1987, Johnson and Jackson, 1987, Syamlal et al., 1993, Johnson et al., 1990, Srivastava and Sundaresan, 2003, Benyahia, 2008, Reuge et al., 2008, Passalacqua and Marmo, 2009). These models manifest in a well known order-zero dependence on the rate of deformation in the quasi-static flow regime (Srivastava and Sundaresan, 2003) and the simple closures for frictional shear stresses prove to be suitable in the dense regime (van Wachem and Almstedt, 2003). Additionally, it is assumed that the normal frictional stresses (Johnson and Jackson, 1987, Syamlal et al., 1993) are simple monotonically increasing functions with increasing volume fraction. Although these models capture the right dependence of the discharge rate from a bin on the diameter of the orifice (Benyahia, 2008) compared with the Beverloo equation (Beverloo et al., 1961) and are proven to deliver constant discharge rates in the inertial regime, they do not provide a correct quantitative measure of the mass flow rate from hoppers and bins.
In contrast to the above simplifying assumption concerning the normal frictional stresses, da Cruz et al. (2005) suggested that firstly, the granular assembly dilates and secondly, the angle of internal friction varies under shear, which is also supported by experiments. Furthermore, the amount of dilation is also a function of the particle diameter and density (da Cruz et al., 2005, Jop et al., 2006, Forterre and Pouliquen, 2008) and determines the normal frictional stresses. For example, when a granular material is moving in a channel, a source of dilation occurs. In order to allow the motion, the material needs to dilate close to the wall, where a shear band develops. In the centre of the channel the material remains unsheared instead and the porosity constant around the random packing limit (Artoni et al., 2011).
In this paper, we present a frictional-kinetic closure for the solids stress tensor for KTGF based multi-fluid models, which is given by the addition of the collisional, kinetic and frictional stresses (Johnson and Jackson, 1987, Srivastava and Sundaresan, 2003). In Section 2.1 the kinetic and collisional stresses are deduced from Agrawal et al. (2001) and Hrenya and Sinclair (1997) incorporating the effects of the interstitial gas and the presence of the bounding walls. In Section 2.2 the frictional stress tensor is discussed, which is written in a non-Newtonian form. Here, the normal stresses are determined by the dilation law of da Cruz et al. (2005) and the shear stresses are derived from a shear rate dependent rheology law (da Cruz et al., 2005, Jop et al., 2006, Forterre and Pouliquen, 2008, Lagrée et al., 2011). The additive treatment of the individual contribution to the solids stress tensor requires a modification of the radial distribution function appearing in KTGF, which is presented in Section 2.2.5. In Section 3 details of the implementation are discussed. Then a twofold validation of the model is presented in Section 5. Firstly, the model is tested against experimental data of van Buijtenen et al. (2011) for multiple-spout pseudo-2D fluidized beds. Secondly, the discharge rates from a rectangular bin are studies for different grain diameters, which are in the order of magnitude of the orifice diameter. The results are compared to velocity profile and mass flow rate measurements. It is shown that the model is able to predict the flow in both cases and the discharge rates from the bin correctly. A conclusion ends this paper.
Section snippets
Model equations
The averaged continuity equation for phase q is written in Eq. (1). Note, Eqs. (1), (2), (3), (14) are summarized in Table 1. In Eq. (1) denotes the velocity, the volume fraction and the density of phase q. In case of mono-disperse gas–particle flows q denotes either the gas phase g or the solid phase s. The averaged momentum equations for the gas–solid flow are given in Eqs. (2), (3). Sensitivity studies (van Wachem et al., 2001) suggest that the drag law of Wen and Yu (1966) well
Implementation
We apply the CFD solver FLUENT (version 14) to solve the model equations (Table 1), whereby Eqs. (6), (7), (8), (9), (10), (12), (13), (14) are not covered by its standard functional range. These are, therefore, implemented by user defined functions. For the discretization of the transport Eqs. (1), (2), (3), (4) a second-order upwind scheme is used. The derivatives appearing in the diffusion terms in Eqs. (2), (3), (4) are computed by a least squares method, and the pressure velocity coupling
Multiple-spout pseudo-2D fluidized beds
This well documented test case was investigated numerically with a CFD-DEM approach and experimentally by van Buijtenen et al. (2011) (Fig. 2). They studied the effect of multiple spouts on the spout fluidized bed dynamics. At different heights time averaged solids velocity profiles within the bed are available from PIV (particle image velocimetry) and PEPT (positron emission particle tracking) measurements. It is, therefore, practical to validate the presented frictional-kinetic model by this
Single-spout pseudo-2D bed
In Fig. 5 the computed time averaged (averaging period of 20 s) vertical solids velocities are shown. In the area of the spout the particulate phase is highly accelerated by the gas flow. While the particle–wall collisions in the off-spout region, do not involve sliding, the accelerated colliding particles may slide depending on the friction coefficient and their impact speeds. Using the boundary conditions of Johnson and Jackson (1987) show an under-estimation of the time averaged vertical
Conclusions
In this paper, we have introduced a new closure for the solids stress tensor for KTGF based multi-fluid models. Following Srivastava and Sundaresan (2003) it is supposed that kinetic, collisional and frictional stresses are additive. However, this assumption requires a modification of the radial distribution function (Section 2.2.5). The frictional stress model is based on two simple constitutive laws for the coefficient of internal friction, , and the volume fraction, (da Cruz et
Nomenclature
Latin symbols a, b dimensions of the exit orifice of the rectangular bin CD drag coefficient (see Eq. (5)) rate of deformation tensor for phase q Dh hydraulic diameter of the exit orifice of the rectangular bin (see Eq. (43)) ds particle diameter es coefficient of restitution for particle–particle collisions ew coefficient of restitution for particle–wall collisions g0 radial distribution function Is inertial number (Eq. (28)) ls mean free path of the particles k parameter in the Beverloo Eq. (44) characteristic
Acknowledgments
This work was funded by the Christian-Doppler Research Association, the Austrian Federal Ministry of Economy, Family and Youth, and the Austrian National Foundation for Research, Technology and Development. The authors want to thank David Schellander (Christian Doppler Laboratory on Particulate Flow Modelling, Linz, Austria), who thoroughly reviewed the manuscript.
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