# Mass, Momentum, and Constitutive Equations

Tables 2.1 and 2.2 summarize the governing equations and constitutive equations for different regimes of fluid-particle flow.

Table 2.1 Two-fluid model governing equations

 Conservation of mass Gas phase C - (eg Pg ) + V.(eg Pgvg ) = m g (T-1) Solid phase C ~(es Ps) + V.(espsvs) = rii s ct (T-2) e + e = 1 g s (T-3) Conservation of momentum Gas phase Ct (g PgVg ) + V' (g PgVgVg )=~eg VP + V'Tg + eg Pgg - Pgs (g - V ) (T-4) Solid phase C f Ct (sPsVs )+V'(es PsVsVs )= -es VP + V -VPs +esPsg + Pgs (g - ) (T-5) Conservation of species Gas phase Ct (eg Pgy, )+V-(eg PgVgy, ) = R (T-6) Solid phase: C Zr(esPsyi ) + V.(es Psvsyi) = Rj ct (T-7) Conservation of solid phase fluctuating energy 2 Cpq} + V^pseseVs) =(-? Д + r,):Vvs +V?(0)-y + fgs (T-8)

Table 2.2 Two-fluid model constitutive equations

 Gas phase stress G — eg mg [vg + (yvg у1- 1 egmg (v • vgу (T-9) Solid phase stress g =em[ +(vvsу]-3,m ]v-vs/ (T-10) Collisional dissipation of solid fluctuating energy r, = 3(i- yp,goe[j-jp (T-11) Radial distribution function g • fe г i-1 (T-12) Solid phase pressure Ps — Pknetc + Pcollslon — SsPA + 2Ps (1 + Pss ^go0s Ps < ?s,fr (T-13) Ps Pkinetic + Pcollision + Pfriction Ps ~ Ps,fr (T-14) ^ (e, -Pmn)q Pcritical — FrA 4p ’ (emax -P )P emin — 0.5 ?es ? emax — 0.63, Fr — 0.1es, q — 2,p — 3 (T-15) f ч1/n—1 P friction i 1 V-Us Pcritical [ W2 sin f s : S + (в/dp2)^ (T-16) Solid phase shear viscosity №s № kin + № col + №fr (T-17) 4 , n jq Pktn — es Psdpg0(1 + e) 5 p (T-18) 10p d fp) Г 4 i2 mcoi — 96(1 + e)gp [1 + 5 g0es(1 + e)J (T-19) 3Xsv.Us -— I 6 . mr — -r==- L sinf Ф] Laux (1998) f 2f3.II 2D (9 - sin2 Ф0 (T-20)

Solid phase bulk viscosity

Table 2.2 (continued)

 „4 в Xs — ~?s Psdpg 0(1 + e)J— e f Lun et al. (1984) 3 p ,f (T-21) e Ps x — - e >e , Dartevelle (2003) ? 4sin2 ф.112D + (VUs ) ' (T-22) Conductivity of the fluctuating energy 150p 6 . л2 2 / в K — 384(1 + e)g0 } + 5 esg0(1 + + 2psesdp(1 + e)g^„ where (T-23) q — KsV в (T-24) Granular energy exchange between phases Laminar flow jgs —-3р^в Gidaspow et al. (1992) (T-25) Disperse turbulent flow jgs — bgs (g2kfgbe - 2kf) Sinclair and Mallo (1998) kf is the turbulent kinetic energy of the gas phase. (T-26) Johnson and Jackson (1987) boundary condition for particles 6Pses,max dVs,w s,w •J3pfps?sg0y[e dx (T-27) q кв 80w 4ьлфра?vl„pg0q3'2 g dx 6e g w s,max w (T-28) g — Sp(~ e2„ )?sPsg0q3/2 w 4?s max (T-29)

Table 2.2 (continued)

 Interphase exchange (drag) coefficient Modified Wen and Yu model (1966) for concentrated or non-homogeneous solid phase: 3 (1 - e )e I b = 4( dsp) gr к -VsKh (T-40) where Hd is the heterogeneity factor (see Chapter 4 for more details) 24 { Rer < 1000; CD0 = (l + 0.15Re“87 ) Re p (T-41) Rep > 1000; CD0 = 0.44 (T-42) „ ^ - Vsdp Re p = p m (T-44) Syamlal and O’Brien drag model (Syamlal et al. 1994) for very dilute or homogeneous solid phase: 4 (1 -e )e i . Re b = 4 dp .v2 Г V - Vs CD0.( ) (T-44) where vts is the particle terminal velocity, vs = 0.5(A - 0.06 Rep + J(0.06 Rep)2 + 0.12Rep(2B - A) + A2 ) (T-45) with A and B having a form of a=<» „„ в085 g К ,e>0.85 (T-46) a and bare the adjustable parameters.