EMMS model

The "variational type of multi-scale CFD, here, refers to CFD with meso-scale models featuring variational stability conditions. This approach can be exemplified by the coupling of the EMMS model (Li and Kwauk, 1994) and TFM, where the EMMS/matrix model (Wang and Li, 2007) at the subgrid level is applied to calculate a structure-dependent drag force. 

In summary, we may expect that the correlative type of multi-scale CFD can be used for the problems with clear scale separations between the micro-scale and the meso-scale, while the variational type, provided with appropriate stability condition, seems free of such limitation. In what follows we will detail some examples of the variational approach by introducing its basis of the EMMS model. 

Energy-minimization multi-scale (EMMS) model—a meso-scale model... 

The EMMS model was first proposed for the hydrodynamics of concurrent-up particle-fluid two-phase flow. Though it is based on a rather simplified physical picture of the complex system (Li, 1987 Li and Kwauk, 1994), it harnesses the most intrinsic complexity in the system, the meso-scale heterogeneity, and this is why it allows better predictions to the critical phenomena in the system which is obscured in other seemingly more comprehensive models. 

To facilitate the discussion in the rest part of this article, we revisit the formulation of the EMMS model, while interested readers are referred to Li and Kwauk (1994) and Li et al. (2005) for more details. 

These six equations are insufficient to give a closure of the EMMS model that involves eight variables. The closure is provided by the most unique part of the EMMS model, that is, the introduction of stability condition to constraint dynamics equations. It is expressed mathematically as Nst = min, which expresses the compromise between the tendency of the fluid to choose an upward path through the particle suspension with least resistance, characterized by Wst = min, and the tendency of the particle to maintain least gravitational potential, characterized by g = min (Li and Kwauk, 1994). 

The EMMS model was proposed for the time-mean behavior of fluidized beds on the reactor scale. A more extensive application of the EMMS model to gas-solid flow is through its coupling with the two-fluid CFD approaches, which brings about an EMMS-based multi-scale CFD framework for gas—solid flow. For this purpose, Yang et al. (2003) introduced an acceleration, a, into the EMMS model to account for the... 

It should be noted that if the inertial terms are omitted, the above relations will return to the original form of the EMMS model (Li and Kwauk, 1994). For specified conditions Us, Gs, and g, this set of 10... 

Figure 13 The apparent flow regime diagram calculated with EMMS-based multiscale CFD and the intrinsic flow regime diagram for the air-FCC system (fluid catalytic cracking particle, dp = 54 m, pp = 930 kg/m3) calculated by using the EMMS model without CFD. The intrinsic flow regime diagram is independent of the riser height (Wang et al., 2008).

Albeit originally proposed for gas-solid fluidization, the concepts of structure resolution and compromise between dominant mechanisms embodied in the EMMS model can be generalized into the so-called variational multi-scale methodology (Li and Kwauk, 2003) and extended to other complex systems (Ge et al., 2007). One typical example out of these extensions is the Dual-Bubble-Size (DBS) model for gas-liquid two-phase flow in bubble columns (Yang et al., 2007, 2010). 

Bridging the gap between micro- and macro-scale is the central theme of the first contribution. The authors show how a so-called Energy-Minimization Multi-Scale (EMMS) model allows to do this for circulating fluid beds. This variational type of Computational Fluid Dynamics (CFD) modeling allows for the resolution of meso-scale structures, that is, those accounting for the particle interactions, and enables almost grid-independent solution of the gas-solids two-phase flow. 

IV. Two-Phase Model—Energy-Minimization Multi-scale 159 (EMMS) Model... 

Then, overall hydrodynamics of fast fluidization—region—will be discussed by extending the EMMS model to both axial and radial directions. Other two aspects of local hydrodynamics—regime and pattern—will not be involved as this book is limited to the fast fluidization regime. 

This is a nonlinear optimization problem with eight parameters and nine constraints, called energy-minimization multi-scale (EMMS) modeling, from which the parameter vector X and various energy consumptions can be calculated. 

For solving the EMMS model, the saturation carrying capacity K has to be determined to ascertain whether Nn should be minimized or maximized in Model LG, that is, to identify the transition from the PFC regime to the FD regime. This transition is characterized by the following equality (Li et al, 1992) ... 

The EMMS model is a nonlinear optimization problem involving eight parameters and nine constraints consisting of both equalities and inequalities,... 

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