Eulerian-Eulerian modelling

In the Eulerian–Eulerian approach, the gas and the solid phases are handled as interpenetrating continua in Eulerian frame. This is also called the two-fluid method (TFM) as the solid phase is treated as a fluid. This is the most commonly used approach for simulating flow dynamics of fluidized bed applications (Myöhänen et al., 2006).

The formulation of the averaged transport equations for two-fluid models is usually credited to Ishii (1975) or Anderson and Jackson (1967). Both of them have derived the flow equations from first principles. Originally, the equations by Ishii were developed for modelling liquid-gas flows in nuclear processes and the equations by Anderson and Jackson for modelling fluidized beds. Van Wachem et al. (2001b) showed that the difference between the two formulations was in the effect of fluid stress tensor on the solid phase and that the Ishii's treatment was appropriate for a dispersed phase consisting of fluid droplets and Anderson and Jackson's treatment was appropriate for dispersed phase consisting of solid particles.

The following presents governing equations for locally averaged variables following the formulations of Anderson and Jackson.

Continuity equations for gas and solids:

· 0 (2.5)

· 0 (2.6)

Momentum equations (gas phase g, solid phases s, f):

· · (2.7)

·

· (2.8)

In some formulations in literature, the solid pressure (Ps) may have been written inside solid phase stress (τs) (Arastoopour, 2001; Gidaspow et al., 2004). The interphase exchange force between multiple solid phases is often neglected because the studies are limited to one solid phase. Moreover, the interphase forces could include lift force and virtual mass force, but these are generally considered insignificant compared with the drag force due to large density difference between the phases. The challenges of the two-fluid models are related to correct definition of the stress terms, solid pressure and interphase drag coefficients (β).

The mostly used approach is to apply kinetic theory concepts for defining the terms due to particle-particle interactions, i.e. the solid phase stress tensor (τs) and solid pressure (Ps); hence, the model approach is named as kinetic theory of granular flow (KTGF).

Bagnold (1954) is credited for starting the kinetic theory approach of granular flow.

Major efforts to the development of KTGF theory and the closure models have been contributed by Ogawa et al. (1980), Jenkins and Savage (1983), Lun et al. (1984), Sinclair and Jackson (1989), Ma and Ahmadi (1990), Ding and Gidaspow (1990), Gidaspow et al. (1992) and Syamlal et al. (1993).

In the granular theory, the analogy with kinetic gas theory is attempted. The kinetic energy related to random movement of solid particles is interpreted as granular temperature θs. On the other hand, the granular temperature can be understood as some kind of turbulent kinetic energy or solids fluctuating energy. The instant particle

velocity v can be thought to be decomposed into a mean velocity and a superimposed fluctuating velocity .

(2.9)

The basic principle of KTGF is that analogous to the thermodynamic temperature for gases, a granular temperature θs can be introduced, which is associated with the random fluctuating velocity of the particles.

1

3 (2.10)

The granular temperature can be solved from a transport equation (Ding and Gidaspow, 1990; Arastoopour, 2001) or by algebraic formulation, in which the convection and diffusion of θs have been neglected (Syamlal et al., 1993). The stress terms are then functions of the granular temperature.

The gas-solid momentum exchange is defined by drag coefficient βgs, which has been empirically determined by different researchers for different conditions. The formulation by Syamlal and O'Brien (1989) applies equations developed by Dalla Valle (1948) and Garside and Al-Dibouni (1977). The formulation by Gidaspow et al. (1992) applies equations by Ergun (1952) for dense flows and equations by Wen and Yu (1966) for dilute flows, but resulting in a step change at solid volume fraction 0.2.

Equations for solid-solid drag term βfs have been proposed by Gidaspow et al. (1986), Syamlal (1987) and Bell (2000).

The different closure models and correlations have been reviewed by van Wachem et al.

(2001b). There are no unique formulations in the literature for the closure models defining the different terms in the momentum equations.

In order to capture the meso-scale flow features, the calculation mesh spacing should be relatively fine, in the order of 10...100 particle diameters (Agrawal et al., 2001;

Andrews et al., 2005). In a typical CFB combustor, the average particle size is in the order of 200...300 µm, which would mean a cell size of 2...30 mm. For a large-scale furnace, this would mean calculation mesh sizes in the order of 10⁹...10¹² elements, which is too demanding for any practical calculations. Consequently, large-scale studies need to be performed with coarse calculation meshes. The clusters smaller than the cell size cannot be resolved, which in coarse mesh leads to overestimating the drag force between the gas and solid phases and false macroscopic flow fields. Thus, the modelled volume fraction of solids tends to be too small at the lower part of the furnace or too high at the upper part of the furnace. Several researchers have addressed this problem by suggesting modifications to the drag term or development of sub-grid models (Agrawal et al., 2001; Zhang and VanderHeyden, 2002; Yang et al., 2003; Andrews et

al., 2005; Kallio, 2005; Qi et al., 2007; Igci et al., 2008). Lately, especially the EMMS (energy minimization multiscale) method, which is based on correcting the drag coefficient, has been extensively used and has succeeded matching the axial solid profiles with the measurements (Wang‚ W. et al., 2010).

The time step of the transient calculations must be sufficiently small to capture fast movements of solid phase and to achieve stable calculation process. Typically, the time steps are in the order of 1 ms. Increasing the time step size and, ultimately, achieving a steady state CFD simulation is an attractive alternative for time-consuming transient simulations. This has been pursued by De Wilde et al. (2007) and Kallio et al. (2008).

Naturally, a steady state macroscopic flow field can be generated by averaging over a transient simulation, but due to long calculation times, the averaging times are often relatively small, in the order 20 s of simulated process time (Shah et al., 2009; Zhang et al., 2010). Considering the possible slow fluctuations of the CFB process, these kind of averaging times may be too small to represent the actual steady state model results and the sensitivity of results on the averaging time should be checked.

Another item to consider when applying averaged CFD calculations is that the effects of transient phenomena on the mixing are lost in the averaging process. These transient phenomena have an effect e.g. on the mixing of reactants and the combustion process.

In the averaged, steady state flow equations, these effects create new terms, analogous to single phase turbulence models, and the challenge is how to determine proper closure models for the new terms.

Due to challenges related to modelling the large-scale CFB processes with two-fluid models, most of the published studies, including quite recent ones, are limited to two-dimensional cases or small-scale applications (Mathiesen et al., 2000; Flour and Boucker, 2002; Yue et al., 2008; Zhang et al., 2008; Hartge et al., 2009; Nikolopoulos et al., 2009; Özel et al., 2009; Wang‚ J., 2010; Wang‚ X.Y. et al., 2010). The published industrial scale 3D CFD studies are very scarce and none of them includes modelling of combustion and heat transfer. Myöhänen et al. (2006) present model results of a 102 MWe CFB, in which a three-dimensional slice of the furnace was modelled, but the simulated process times were very short. Shah et al. (2009) show calculation of a full furnace with two solid phases to better simulate the measured vertical pressure profile.

Zhang et al. (2010) performed a simulation of a 150 MWe boiler modelling a full CFB loop including two cyclones and the return leg system. The simulated process time was 40 seconds. The results compared well with the measurements, when the interphase drag term had been modified based on the EMMS method.

Many researchers are working together with boiler industry on improving the applicability of the TFM models for practical calculations of CFB combustors. In near future, the number of large-scale studies will certainly increase due to improvement of calculation capacity, numerical methods, model theories, and, especially, with support of advanced measurement techniques for validating the models.