mfixequations2005-4-4

MFIX Equations 2005-4-4 (August 2009) 1/16

Summary of MFIX Equations

Table of Contents A. Governing equations ................................................................................................................................. 2 B. Kinetic Theory ........................................................................................................................................... 3

Constitutive equations ................................................................................................................................ 3 Algebraic granular energy equation ........................................................................................................... 5

C. Frictional Stress Models ............................................................................................................................ 5 Schaeffer model ......................................................................................................................................... 5 Princeton model ......................................................................................................................................... 6

D. Interface Momentum Transfer .................................................................................................................. 7 Wen-Yu drag correlation ........................................................................................................................... 7 Gidaspow drag correlation ......................................................................................................................... 8 Hill-Koch-Ladd drag correlation ............................................................................................................... 8 Syamlal and O’Brien .................................................................................................................................. 9 Solids/solids momentum exchange coefficient ........................................................................................ 10

E. Correlations for maximum packing ......................................................................................................... 10 Yu-Standish correlation ........................................................................................................................... 10 Fedors-Landel correlation ........................................................................................................................ 11

F. Gas momentum equation constitutive models ......................................................................................... 12 Stresses ..................................................................................................................................................... 12 Porous media model ................................................................................................................................. 12

G. Gas/Solids Turbulence models ................................................................................................................ 12 H. Energy equation constitutive models ...................................................................................................... 12

Interphase heat transfer ............................................................................................................................ 12 Gas and solids conduction ....................................................................................................................... 13 Heats of reaction ...................................................................................................................................... 13

Nomenclature ............................................................................................................................................... 14 References .................................................................................................................................................... 16

Refer to this document as: S. Benyahia, M. Syamlal, T.J. O’Brien, “Summary of MFIX Equations 2005-4”, From URL https://mfix.netl.doe.gov/documentation/MFIXEquations2005-4-4.pdf , August 2008.

Modifications to the previous version https://mfix.netl.doe.gov/documentation/MFIXEquations2005-4-3.pdf

• Corrected Equation A3 and A4. The MFIX code contains mass-transfer source terms due to subtraction of continuity equations from conservative form of A3-A4.


The purpose of this document is to summarize the current set of equations in MFIX. This document will be updated when the equations in MFIX are revised or errors in this document needs to be fixed. The equations are listed here without any explanation, to expedite the publication of this document. Some details about the equations may be found in the two previous MFIX documents [1, 2]; be aware that some of the equations in those documents have been revised. Refer to the readme file for the keywords (to be used to set up an MFIX simulation) for selecting the different equation choices presented here. A. Governing equations Einstein summation convention implied only on subscripts i and j. Continuity equations for solids phases m = 1, M:

( ) ( ) RUxt mn

N

1=nmimm

imm

m

∑=∂∂

+∂∂ ρερε (A1)

Continuity equation for gas phase g:

( ) ( ) RUxt gn

N

=1ngigg

igg

g

∑=∂∂

+∂∂ ρερε (A2)

Momentum equations for solids phases m = 1, M:

( ) ( )

imm

M

kkmigmi

j

mij

i

gmmimjmm

jmimm

g

IIxx

PUU

xU

t

ρε

τερερε

+

−+∂

∂+

∂

∂−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

+∂∂

∑=1 (A3)

Momentum equations for gas phase g:

( ) ( )

igg

giM

mgmi

j

gij

i

gggigjgg

jgigg

g

fIxx

PUU

xU

t

ρε

τερερε

+

+−∂

∂+

∂

∂−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

+∂∂

∑=1 (A4)

Granular temperature equations for solids phases m = 1, M

mmmmj

mimij

i

mm

ij

mmjmmmm J

xU

xxxU

tρετκ

εερ −∏+∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂Θ∂

∂∂

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

Θ∂+

∂Θ∂

23

(A5)

Energy balance equations for solids phases m = 1, M

( ) ( )44mRmRmmgmgm

i

mi

j

mmj

mpmmm TTHTT

xq

xTU

tTC −+Δ−−−

∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

+∂∂ γγρε (A6)

Energy balance equation for gas phase g:


( ) ( )44

1gRgRgg

M

mgmgm

i

gi

j

ggj

gpggg TTHTT

xq

xT

Ut

TC −+Δ−−+

∂

∂−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+

∂

∂∑=

γγρε (A7)

Species balance equations for solids phases m = 1, M

( ) ( ) RxXD

xXU

xX

t mni

mnmn

imnmimm

imnmm +⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂∂∂

=∂∂

+∂∂ ρερε (A8)

Species balance equation for gas phase g:

( ) ( ) RxX

Dx

XUx

Xt gn

i

gngn

igngigg

igngg +⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂

∂∂

=∂∂

+∂∂ ρερε (A9)

B. Kinetic Theory Constitutive equations This is a modified Princeton model [3]. Modifications include the ad-hoc extension of kinetic theory to polydisperse systems (more than one solids phase), which guarantees that two identical solids phases will behave same as one solids phase. Solids stresses:

mijmiji

mibmmij S

xU

P μδημτ 2+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+−= (B1)

where

i

mi

i

mj

j

mimij x

Ux

Ux

US∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=31

21

(B2)

Solids pressure:

( )⎥⎦

⎤⎢⎣

⎡+Θ= ∑

=

M

nmnnmmmm gP

1,041 εηρε (B3)

Solids viscosity:

( ) ( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠

⎞⎜⎝

⎛−+⎟

⎠

⎞⎜⎝

⎛+

−⎟⎠⎞

⎜⎝⎛ +

= ∑∑==

b

M

nmnn

M

nmnn

mm

mm gg

gημεηηεη

ηημαμ

5323

581

581

232

1,0

1,0

,0

*

(B4)


( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+Θ

Θ=

∑= mm

M

nmmnnm

mmmmmm

g

g

ερβμερ

μερμ

21

,0

,0* (B5)

mpmd Θ= πρμ965

(B6)

( )∑=

=M

nmnnmb g

1,05

256 εμεπ

μ (B7)

Solids conductivity:

( ) ( ) ( )

( ) ( )⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠

⎞⎜⎝

⎛−+

⎟⎠

⎞⎜⎝

⎛−+⎟

⎠

⎞⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛=

∑

∑∑

=

==

2

1,0

2

1,0

2

1,0

,0

*

33412564

345

1215

121

M

nmnn

M

nmnn

M

nmnn

mm

mm

g

gg

gεηη

π

εηηεηκκ (B8)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+Θ

Θ=

∑= mm

M

nmmnnm

mmmmmm

g

g

ερβκερ

κερκ

56

1,0

,0* (B9)

( )ηηπρ

κ334148

75−

Θ= mpmd

(B10)

Collisional dissipation:

( )( )

2/31,0

148m

p

M

nmnn

m d

gJ Θ−=

∑=

εηη

π (B11)

21 e+

=η (B12)

Exchange terms:


mmpmm

mggmsm dg Θ

−+Θ−=Π

πρ

μεβ

3,0

22813

uu (B13)

Algebraic granular energy equation MFIX offers an option to solve algebraic granular energy equation, which is derived by equating the production to dissipation. Note that this is equation was revised in 2005.

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

ΘK2

)]D(DK2 + (DK[K4 + (DK + DK- =

4mm

mijmij3mmii2mm4m2mmii

21mmiim1m

2

mε

εεε ))) 22

(B14)

g )e + (1 2 = K 0mmm1m mmρ (B15)

K32 - )(3 / g )e+(1 d 4 = K 3m0mmmmpm2m mm

περ (B16)

( )⎭⎬⎫

⎩⎨⎧

+π

εε

πρ5

)e+(1g8 + ]g1)-e)(3e+0.4(1 + 1e3[

)e-3(3

2d = K

mm0m

0mmmmmmmmm

mpm3m

mm

mm5.0

(B17)

πρ

dg )e - 12(1

= Kpm

0mmm2mm

4m (B18)

C. Frictional Stress Models Schaeffer model This model [4] is used at the critical state when the solids volume fraction exceeds the maximum packing limit.

( )⎪⎩

⎪⎨⎧

≥

<−=

*

*10*24

0

10

εε

εεεε

g

ggcP (C1)

(Note that the constant in the code is 1025 dyne/cm2).


( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≥

<

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

= ∑=

*

*max

1

2

0

,4sinmin

εε

εεμε

εφμ

g

gsM

mm

s

D

c

f IP

(C2)

100max =sμ (C3)

(Note that this constant in the code is 1000 poise).

0=bulkfμ (C4)

( ) ( ) ( )[ ] 231,

223,

212,

211,33,

233,22,

222,11,2 6

1sssssssssD DDDDDDDDDI +++−+−+−= (C5)

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂=

i

js

j

isijs x

uxu

D ,,, 2

1 (C6)

Princeton model This model [5] is a modification of Savage model that accounts for strain-rate fluctuations. Also the frictional model influences the flow behavior at solids volume fractions below maximum packing ( 5.0min =sfε ).

( )( )( )( ) ( )

( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−≥

−<≤−

−−

<−

=

min

min*

*

min

*10*24

10

11

10

sfg

sfgsg

rsg

gg

c FrP

εε

εεεεε

εε

εεεε

(C7)

Where Fr =0.05, r= 2, s=5. (Note that the constants in the code are 0.5 and 1025 dyne/cm2).

( ) ∑=

−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Θ+

⋅∇−= M

mm

s

n

pc

f

dnPP

1

1

2/:sin21

ε

ε

φ SSv

(C8)


( ) ( )∑=

−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

Θ+= M

mm

sn

c

f

p

ff P

Pnn

d

P

1

11

21

/:

sin2

ε

εφμ

SS (C9)

Here, the coefficient n is set differently depending on whether the granular assembly experiences a dilatation or compaction:

( )⎪⎩

⎪⎨

⎧

<⋅∇

≥⋅∇=

003.1

0sin2

3

v

vφn (C10)

fbulkf μμ

32

−= (C11)

D. Interface Momentum Transfer Gas/solids momentum interface exchange:

( )migigmgmi uuI −= β (D1) Solids/solids momentum exchange: ( )mikikmkmi uuI −= β (D2) Wen-Yu drag correlation

65.2

43 −

−= g

pm

mgmggDgm d

C εεερ

βuu

(D3)

( )

⎩⎨⎧

≥<+

=1000Re44.01000ReRe15.01Re/24 687.0

DC (D4)

g

pmmggg dμ

ερ uu −=Re (D5)


Gidaspow drag correlation

( )⎪⎪

⎩

⎪⎪

⎨

⎧

<−

+−

≥−

=

−

8.075.11150

8.043

2

65.2

gpm

mgmg

pmg

ggs

ggpm

mgmggD

gm

dd

dC

εερ

εμεε

εεεερ

βuu

uu

(D6)

( )

⎩⎨⎧

≥<+

=1000Re44.01000ReRe15.01Re/24 687.0

DC (D7)

g

pmmggg dμ

ερ uu −=Re (D8)

Hill-Koch-Ladd drag correlation (valid for one solids phase only) The drag correlation of Hill, Koch and Ladd [6, 7] was modified [12] and implemented in MFIX.

( ) 22118

pmmmggm d

Fεεμβ −= (D9)

The drag force ( F ) is given as:

( )( )3

2

8/31

Re01.0Re8/31F

FandF s −

−≤≤+= ε (D10)

( )1

2012

33210 2

4Re01.0

FFFFFF

andRFFF se

−−+≤>+= ε (D11)

( )

( )( )

⎪⎪

⎩

⎪⎪

⎨

⎧

−−+>>

−−

>≤

+=

1

2012

33

3

2

32

24

Re01.0

8/31

Re01.0

Re

FFFFFF

and

FF

and

FFF

s

s

ε

ε

(D12)

And the coefficients are defined as follows:


( ) ( ) ( )( )

( )⎪⎪

⎩

⎪⎪

⎨

⎧

≥−

<<⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−+

+++−

=

4.01

10

4.001.01

1016.848.8681.01

14.17ln64/1352/311

3

332

0

ss

s

ss

s

sss

ssss wwF

εεε

εεε

εεε

εεεε

(D13)

⎪⎩

⎪⎨

⎧

>+

≤<=

1.0)6.11exp(00051.011.0

1.001.040/2

1

ss

ssF

εε

εε (D14)

( ) ( ) ( )( )

( )⎪⎪

⎩

⎪⎪

⎨

⎧

≥−

<⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−+

+++−

=

4.01

10

4.01

1041.1503.11681.01

89.17ln64/1352/311

3

332

2

ss

s

ss

s

sss

ssss wwF

εεε

εεε

εεε

εεεε

(D15)

( )⎪⎩

⎪⎨⎧

≥−++

<+=

0953.01/0232.0212.00673.0

0953.003667.09351.053

sss

ssFεεε

εε (D16)

( )g

pmmgmg dμ

ερ

21

Reuu −−

= (D17)

( )( )ssew εε /4.010 −−= (D18)

Syamlal and O’Brien

( ) mgmrm2

pm2rm

ggmgm Re/V4.8 + 0.63

dV43

= uu −ρεε

β (D19)

( ) A + A)-(2B Re 0.12 + )Re (0.06 + Re 0.06 - A 0.5 = V 2m

2mmrm (D20)


ε 4.14g = A (D21)

⎪⎩

⎪⎨⎧ ≤

0.85 > if

0.85 if 0.8 = B

g2.65g

g1.28g

εε

εε (D22)

μ

ρ

g

gmgpmm

d = Re

uu − (D23)

Solids/solids momentum exchange coefficient

( ) ( )( ) ccoefkmmkkm

kpkmpm

kpmpfkmkm Psg

dddd

ce +−+

+⎟⎟⎠

⎞⎜⎜⎝

⎛++= uu,03

,3

,

2,,

2

28213 ρρ

ρρπππβ (D19)

fkmc : Constant defined in input file (no default value assigned)

coefs : Constant defined in input file with default value of zero. (See reference [10] and [11] for details) E. Correlations for maximum packing This section provides description of two empirical correlations for computing the solids maximum packing in polydisperse systems by Yu and Standish [8] and Fedors and Landel [9]. To use these correlations, the numbering of the solids phases was rearranged in MFIX to start with the coarsest to the finest powder. Yu-Standish correlation This correlation can be used for powder mixtures with 2 or more components.

max,

* 1 mixturesεε −= (E1)

Mi

Xcx

pXcx

p

M

ij ij

i

ij

isi

j ij

i

ij

is

ismixtures ...,,2,1

111min

1

max,

1

1

max,

max,max

, =

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟

⎟⎠

⎞⎜⎜⎝

⎛−−

=

∑∑+=

−

=

εε

εε

(E2)


∑=

= M

jjs

isicx

1,

,

ε

ε (E3)

⎪⎪

⎩

⎪⎪

⎨

⎧

≥−

−−

<−

−

=

ijr

ijr

X

is

ij

is

ij

ij

max,

2

max,

2

21

1

21

ε

ε (E4)

( )( )⎪⎩

⎪⎨⎧

>

≤+−−+=

741.0

741.035.135.211max,

2max,

max,

max,

ijis

ijijijisisisij r

rrrp

ε

εεε (E5)

⎪⎪

⎩

⎪⎪

⎨

⎧

<

≥

=

jidd

jidd

r

ip

jp

jp

ip

ij

,

,

,

,

(E6)

Mid ipis ...,,2,1, ,

max, =ε represent the maximum packing and particle diameter of individual powders.

Fedors-Landel correlation This correlation can only be used for a binary mixture of powders.

( ) ( )( )[ ]( )[ ] ( )

( ) ( )[ ] ( )⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

−+>+−+−

−+≤+−+

×−−+−

=

max2,

max1,

max1,

max1,

1max

1,2max

2,max

1,max

1,1,2

max2,

max1,

max1,

max1,

1max

2,max1,

1max2,

max1,

max1,

max2,

max1,1,2

max2,

max1,

max,

111

11

11

sss

sssss

sss

ss

ssss

ssss

mixtures

cxcxr

cxcx

r

εεεε

εεεε

εεεε

εε

εεε

εεεε

ε

(E7)


F. Gas momentum equation constitutive models Stresses

gijgtgij Sμτ 2= (F1) where

iji

gi

i

gj

j

gigij x

Ux

Ux

US δ

∂

∂−⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂=

31

21

(F2)

),( max eggt Min μμμμ += (F3)

Dgggse Il 222 ρεμ = (F4)

( ) ( ) ( )[ ]2

31,2

23,2

12,

211,33,

233,22,

222,11,2 6

1

ggg

ggsgggDg

DDD

DDDDDDI

+++

−+−+−= (F5)

Porous media model

( ) gigjgjggig

gi UUUcUc

f ρμ

22

1

−−= (F6)

G. Gas/Solids Turbulence models The gas/solids turbulence models are given in Benyahia 2005. The equations are not reproduced here because of a small inconsistency in the notation. H. Energy equation constitutive models Interphase heat transfer

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟⎠

⎞⎜⎜⎝

⎛=

1exp 0gm

gmpg

gmpggm RC

RC

γ

γ (H1)

20 6

m

mmggm d

Nuεκγ = (H2)

( )( ) ( ) 3/17.023/12.02 PrRe2.14.233.1PrRe7.015107 mggmggmNu εεεε +−+++−= (H3)


Gas and solids conduction

i

mmmi x

Tq

∂

∂−= κ (H4)

i

gggi x

Tq

∂

∂−= κ (H5)

Heats of reaction

( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+≈

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂

∂

⎥⎥⎦

⎤

⎢⎢⎣

⎡+=Δ−

′′∑∑ ∫

∑ ∫

RXRdTTCH

xX

Ut

XdTTCHH

nm

N

1=nmnmn

n

T

Tpmnnrefm

i

gngi

gngg

n

T

Tpmnnrefmm

ms

ref

s

ref

)(

)(

,

, ρε

(H6)

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+≈Δ− ′

′∑∑ ∫ RXRdTTCHH ng

N

1=ngngn

n

T

Tpgnnrefgg

gg

ref

)(, (H7)


Nomenclature c1 Permeability of porous media; m2 c2 Inertial resistance factor of porous media; m-1 Cpg Specific heat of the fluid phase; J/kg·K Cfkm Coefficient of friction for solids phases k and m Cpm Specific heat of the mth solids phase; J/kg·K dpm Diameter of the particles constituting the mth solids phase; m

gijD Rate of strain tensor, fluid phase; s-1

mijD Rate of strain tensor, solids phase-m; s-1 Dgn Diffusion coefficient of nth gas-phase species, kg/m·s Dmn Diffusion coefficient of nth solids-phase-m species-n, kg/m·s ekm Coefficient of restitution for the collisions of mth and kth solids phases fgi Fluid flow resistance due to porous media; N/m3 gi Acceleration due to gravity; m/s2 g0m Radial distribution function at contact (Hm,ref )n Enthalpy of mth solids phase, species n at Tref; J/m3 (Hg,ref )n Enthalpy of fluid phase, species n at Tref; J/m3 ΔHg Heat of reaction in the fluid phase; J/m3·s ΔHm Heat of reaction in the mth solids phase; J/m3·s i, j Indices to identify vector and tensor components; summation convention is used

only for these indices. I2Dg Second invariant of the deviator of the strain rate tensor for gas phase; s-2 I2Ds Second invariant of the deviator of the strain rate tensor for solids phase 1; s-2 kg Fluid-phase conductivity; J/m·K·s kpm Conductivity of material that constitutes solids phase m; J/m·K·s ksm Solids phase m conductivity; J/m·K·s ls A turbulence length-scale parameter; m m Index of the mth solids phase. "m=0" indicates fluid phase M Total number of solids phases Mw Average molecular weight of gas n Index of the nth chemical species Ng Total number of fluid-phase chemical species Nm Total number of solids phase m chemical species Num Nusselt number Pg Pressure in the fluid phase; Pa Pm

p Pressure in Solids phase m, plastic regime; Pa Pm

v Pressure in Solids phase m, viscous regime; Pa Pr Prandtl number qgi Fluid-phase conductive heat flux; J/m2·s qmi Solids-phase m conductive heat flux; J/m2·s R Universal gas constant; Pa·m3/kmol·K Rem mth solids phase particle Reynolds number Rkm Ratio of solids to fluid conductivity Rmk Rate of transfer of mass from mth phase to kth phase. k or m = 0 indicates fluid

phase; kg/m3·s


Rgn Rate of production of the nth chemical species in the fluid phase; kg/m3·s Rmn Rate of production of the nth chemical species in the mth solids phase; kg/m3·s t Time; s Tg Thermodynamic temperature of the fluid phase; K Tm Thermodynamic temperature of the solids phase m; K TRef Reference temperature; K TRg Fluid phase radiation temperature; K TRm Solids phase-m radiation temperature; K Ugi Fluid-phase velocity vector; m/s Umi mth solids-phase velocity vector; m/s xi ith Coordinate Direction; m Xgn Mass fraction of the nth chemical species in the fluid phase Xmn Mass fraction of the nth chemical species in the mth solids phase GREEK LETTERS βgm Coefficient for the interphase force between the fluid phase and the mth solids

phase; kg/m3·s βkm Coefficient for the interphase force between the kth solids phase and the mth solids

phase; kg/m3·s gmγ Fluid-solids heat transfer coefficient corrected for interphase mass transfer; J/m3·K·s 0gmγ Fluid-solids heat transfer coefficient not corrected for interphase mass transfer;

J/m3·K·s γRg Fluid-phase radiative heat transfer coefficient; J/m3·K4·s γRm Solids-phase-m radiative heat transfer coefficient; J/m3·K4·s Granular energy dissipation due to inelastic collisions; J/m3·s εg Volume fraction of the fluid phase (void fraction) Packed-bed (maximum) solids volume fraction εm Volume fraction of the mth solids phase η Function of restitution coefficient Θm Granular temperature of phase m; m2/s2 λrm Solids conductivity function λv

m Second coefficient of solids viscosity, viscous regime; kg/m·s μe Eddy viscosity of the fluid phase; kg/m·s μg Molecular viscosity of the fluid phase; kg/m·s μgmax Maximum value of the turbulent viscosity of the fluid phase; kg/m·s μgt Turbulent viscosity of the fluid phase; kg/m·s μp

m Solids viscosity, plastic regime; kg/m·s μv

m Solids viscosity, viscous regime; kg/m·s ξmk ξmk = 1 if Rmk < 0; else ξmk = 0. ρg Microscopic (material) density of the fluid phase; kg/m3 ρm Microscopic (material) density of the mth solids phase; kg/m3

gijτ Fluid-phase stress tensor; Pa

mijτ Solids phase m stress tensor; Pa φ Angle of internal friction, also used as general scalar φk Contact area fraction in solids conductivity model


References

1. Syamlal, M., W.A. Rogers, and T.J. O'Brien, 1993. "MFIX Documentation, Theory Guide," Technical Note, DOE/METC-94/1004, NTIS/DE94000087, National Technical Information Service, Springfield, VA.

2. Syamlal, M. December 1998. MFIX Documentation: Numerical Techniques. DOE/MC-31346-5824. NTIS/DE98002029. National Technical Information Service, Springfield, VA

3. Agrawal, K., Loezos, P.N., Syamlal, M and Sundaresan, S., 2001. J. Fluid. Mech., 445, 151-185. 4. Schaeffer, D.G., 1987. J. Diff. Eq., 66, 19-50. 5. Srivastava, A. and Sundaresan, S., 2003. Powder Tech., 129, 72-85. 6. Hill, R.J., Koch, D.L. and Ladd, J.C., 2001. J. Fluid Mech., 448, 213-241. 7. Hill, R.J., Koch, D.L. and Ladd, J.C., 2001. J. Fluid Mech., 448, 243-278. 8. Yu, A.B. and Standish N., 1987. Powder Tech., 52, 233-241. 9. Fedors, R.F. and Landel R.F., 1979. Powder Tech., 23, 225-231. 10. Gera, D., Syamlal M, O'Brien TJ, 2002. Int. J. Multiphase flow, 30 (4), 419-428. 11. Syamlal, M., 1987, "The Particle-Particle Drag Term in a Multiparticle Model of Fluidization,"

Topical Report, DOE/MC/21353-2373, NTIS/DE87006500, National Technical Information Service, Springfield, VA.

12. Benyahia, S., Syamlal M and O'Brien TJ, 2006. Powder Tech., 162, 166-174.

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