mfixequations2005-4-4
TRANSCRIPT
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.
MFIX Equations 2005-4-4 (August 2009) 2/16
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:
MFIX Equations 2005-4-4 (August 2009) 3/16
( ) ( )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)
MFIX Equations 2005-4-4 (August 2009) 4/16
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛+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:
MFIX Equations 2005-4-4 (August 2009) 5/16
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).
MFIX Equations 2005-4-4 (August 2009) 6/16
( )
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≥
<
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
= ∑=
*
*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)
MFIX Equations 2005-4-4 (August 2009) 7/16
( ) ( )∑=
−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
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)
MFIX Equations 2005-4-4 (August 2009) 8/16
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:
MFIX Equations 2005-4-4 (August 2009) 9/16
( ) ( ) ( )( )
( )⎪⎪
⎩
⎪⎪
⎨
⎧
≥−
<<⎥⎥⎦
⎤
⎢⎢⎣
⎡
−+
⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−+
+++−
=
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)
MFIX Equations 2005-4-4 (August 2009) 10/16
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)
MFIX Equations 2005-4-4 (August 2009) 11/16
∑=
= 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)
MFIX Equations 2005-4-4 (August 2009) 12/16
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)
MFIX Equations 2005-4-4 (August 2009) 13/16
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)
MFIX Equations 2005-4-4 (August 2009) 14/16
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
MFIX Equations 2005-4-4 (August 2009) 15/16
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
MFIX Equations 2005-4-4 (August 2009) 16/16
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.