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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)Je 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.

 
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