Energy Minimization MultiScale (EMMS) Approach
The energy minimization multiscale (EMMS) approach was first proposed by Li and Kwauk (1994) based on the coexistence of both dense and dilute regions in a CFB reactor. The model parameters are found by minimization of the massspecific energy consumption for suspending and transporting the particles as the stability criteria for flow structure inside the reactor (Benyahia 2012a).
The EMMS model is able to account for heterogeneous solid structures and cluster formation in the system. Benyahia (2012a) concluded that use of the EMMSbased drag model is accurate and necessary for the prediction of the averaged solid mass and pressure profile along the fully developed flow region of the riser.
Equations (T1) through (T9) in Table 3.1 show the EMMS drag model based on Wang et al. (2008) and Nikolopoulos et al. (2010). In this approach, the Wen and Yu (1966) drag model is multiplied by a heterogeneity factor, H_{d}, that is calculated by solving Eqs. (T1) through (T9) for minimization of Eq. (T10). This is a nonlinear optimization problem that should be solved for any combination of u_{g}, u_{s}, and e_{g}. Finally, the heterogeneity factor is calculated by Eq. (T12) and is implemented in the computational fluid dynamics (CFD) code. Then the drag expression can be expressed as
Table 3.2 presents the closure terms for the EMMS equations.
Table 3.1 EMMS governing equations
Force balance for clusters in unit volume of suspension: 

_{4} Cdc PgV_{S}cv_{sc} + _{4}Qi _{dci}PgV_{si}v_{si}  /(1 ?c)(ps Pg)i.9 ^c) 
(Tl) 
Force balance for dilute phase in unit volume of suspension: 

;^{c}«v _{dp} ^{f} Pg^{v}sfv_{S}f  e_{f})(p_{s} Pg)(g + a_{f}) 
(T2) 
Pressure drop balance between clusters and dilute phase: 

^{C}dc Pg^{v}sc^{v}sc — _{dc[}Pg^{v}sl^{v}si + Cdf _{dp} Ps^{v}s/I^{v}s/I 
(T3) 
Mass conservation for fluid: 

V/(l  /) + v_{c}f = v_{s}?_{e} 
(T4) 
Mass conservation for particles: 

VpfO~ f)+v_{pc}f = V_{p}?_{p} 
(T5) 
Definition of mean voidage: 

a_{g} =fe_{c} + (!“/)?/ 
(T6) 
Definition of voidage inside clusters: 

e_{c} Sg TiOft, Ti 2 
(T7) 
Dense phase voidage standard deviation: 

. 1 (l?p)^{4 }apj l+4e_{p}+4e?4 e+e? 
(T8) 
Mean cluster diameter: 

^{d}ci = d_{p} + (0.027  10d_{p})e_{p} + 32 
(T9) 
Stability condition: 

=  _{(1}__{ae)Pp} ЦF/t;/ + m_{c}F_{c}v_{c} +m_{i}F_{i}v_{f}(l /)]  > min 
(T10) 
Effective drag force: 

Femms = ?_{й}[/( 1 “ ?_{c})(5 + «с) + (1  /)(1  ?/)(fl + ?/)](p_{s}  Pg) tf_{d}(?„) = ^{F}™^{M5} ^{9} Pwen & Fu 
(Tll) (Tl 2) 
Table 3.2 Closure terms for the EMMS model
Dense phase 
Dilute phase 
Interphase 

Effective drag coefficient 
Cdc = ^{c}d0c^{e}^^{4}'^{65} 
^{C}df = ^{c}d0f ^{S}/^{4}'^{65} 
Cdi = cd0i(1  f )^{4}'^{65} 
Standard drag coefficient 
24 3.6 ^{Cd0c} = Re_{c} ^{+} Re°^{313} 
24 3.6 ^{Cd0f} = Re_{f} ^{+} Re_{f}^{0}'^{313} 
24 3.6 ^{Cd0i} = Re, ^{+} Re^{0}'^{313} 
Reynolds number 
Rec = Ы ^{F}g 
Ref = PT^fl Fg 
Re = ^{P}f^{L}lvfl Fg 
Slip velocity 
^{S}c^{v}pc Vsc = Vc . 1  Sc 
_{v v} ^{S}f^{V}Pf ^{Vsf} = ^{V} 1 f 
^{v}s, = ^{(1 f})(v^{f} 1 ^{f}^{P}Sc) 
Drag force 
ndp p_{g} Fc = Cdc 4p yVsdVsd 
Ff = Cdf ^{П}4^{? } Pgvsflvsfl 
ndp p_{g} F, = Cd^ ^{t}^{g}v_{sl}v_{sl} 
Number of particles in cluster 
^{f} 0 ^{} Sc) mc 3 ^{c} *dl 6 
(1  f )(1  Sf) ^{m}f = п^3 6 
f m,, 3 ^{nd}c 6 
Fig 3.1 Heterogeneity factor (H_{d}) as a function of voidage at different slip velocities (This figure was originally published in Powder Technol 288, 2016 and has been reused with permission)
Ghadirian and Arastoopour (2016) calculated the heterogeneity factor H_{d} (the ratio of drag force for nonhomogeneous solidphase flow using the EMMS approach to drag force calculated using the Wen and Yu drag expression for a homogeneous solidphase flow system) as a function of voidage at different specific slip velocities for flow of gas and particles with 185 pm diameter and 2500 kg/m^{3 }density. Figure 3.1 shows the calculated heterogeneity factor H_{d} at slip velocities of 0.5, 1, and 2 m/s. As this figure shows, in very dilute regions of the system, the solid flow pattern approaches toward homogeneous flow. At regions with a solid volume fraction of less than 0.1, H_{d} initially decreases sharply, and then, for a wide range of solid volume fraction greater than 0.15, it levels off at a value of about 0.02 for slip velocities between 0.5 and 2 m/s. This makes the EMMS calculated drag force for the nonhomogeneous solid phase significantly lower than the prediction of any homogeneous drag model. The sudden decrease in H_{d} is because of the presence of clusters that allow the gas to bypass the solids and therefore results in a considerable decrease in the drag force between phases. This figure also suggests that variations in H_{d} with respect to the solid volume fraction are more significant than H_{d }variations with respect to the slip velocity. Therefore, we may neglect the effect of slip velocity variation in most of the gassolid flow systems.
Ghadirian and Arastoopour (2016) simulated bed expansion using 2D TFM CFD equations for both homogeneous and nonhomogeneous particle phases. They concluded that using a nonhomogeneous drag expression, such as EMMS, predicts the bed expansion with noticeably higher accuracy (20 % or less), while homogeneous models used in their study continued to overpredict the bed expansion by up to about 70 % in comparison with the correlation developed based on the experimental data of Krishna (2013).
Figure 3.2 shows the bed expansion factor (final bed height/initial bed height) for the EMMS (developed for 185 qm and 2500 kg/m^{3} density particles) and two homogeneous models as well as the experimentally based correlation of Krishna (2013). To demonstrate the effect of particle type (particle size and density) on the heterogeneity of the system, the results of another set of simulations using the EMMS approach derived for FCC particles (Lu et al. 2009) are also shown in this
Fig. 3.2 Comparison of bed expansion factor as a function of inlet gas velocity using different drag models with experimental data (This figure was originally published in Powder Technol 288, 2016 and has been reused with permission) figure. In the latter case, the EMMS approach was derived by Lu et al. (2009) for FCC particles (75 pm and 1500 kg/m^{3}), but the resulting heterogeneity factor is used to simulate 185 pm bed expansion and 2500 kg/m^{3} particle density.
Figure 3.2 also shows that the homogeneous models predict a very high value for bed expansion with about 70 % deviation from the experimental correlation. Using the EMMS model, the bed expansion factor shows only less than 10% deviation from the experimental values that could be within the experimental error. This graph also shows that the bed expansion calculated based on the EMMS approach derived for FCC particles improves the bed expansion predictions compared to the homogeneous model. It predicts experimental values within a 20 % deviation.