NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Chapter 4 Gas - Solid Two-Phase Flow
Examples,pulverized coal
soot
dust
ash
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
§ 1 Characteristics of Gas-solid Two Phase Flow
Rosin-Pammler Formula
? ? ? ?? ?nkk dddR /e x p ?? (4-1)
R(dk) - weight percent of particle with d > dk
n - index of non-evenness
d - characteristic diameter
? ? ? ? ? ?? ?
n
k
n
kn
k
ddd
d
n
dd
dR /e x p
)(
)1( ?? ?(4-2)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Particle average dimension:
radii average d10 = Snkdk/Snk (4-3)
area average d20 = (Snkdk2/Snk)1/2 (4-4)
volume average d30 = (Snkdk3/Snk)1/3 (4-5)
Sauter average d32 = Snkdk3/(Snkdk2) (4-6)
Typical particle size:
coal in fluidized beds 0 - 10 mm
pulverized coal 1 - 100mm
soot 1 - 10 mm
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2,Density
?m - combined density,?m = Mm/Vm = (Mp + Mf)/Vm
?f - superficial density of fluid,?f = Mf /Vm
?p,?k - superficial density of particle,?p = Mp/ Vm
?p - real density of particle
relation,
? ?? ??????? pkkfkfpfm dn ???????? 6/3
(4-7)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
3,Concentration
Volume concentration
f
p
v
pf
p
v
V
V
C
VV
V
C
?
?
?
(4-8)
Mass concentration
f
p
w
pf
p
w
M
M
C
MM
M
C
?
?
?
(4-9)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
In general,
wv
pm
CC ??
? ???
In pulverized coal transportation
38.0,26.0
1
10,5.0
m i n
3
??
???
??
?
r a n
v
m
f
vw
C
V
V
CC
??
?
(4-10)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
4,Viscosity
A,Einstein:
2)1(
5.21
v
v
f
m
C
C
?
??
m
m (4-11)
When Cv is not high
vfm C5.21 ??m
m
Cv = 10 -20%,F,Guth
21.145.21
vv
f
m CC ???mm
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Cv >20%,D,G,Thomas,
vCvv
f
m eCC 6.162 0275.005.105.21 ????mm
5,Specific heat and conductivity
)1(
)1(
wvfwvpcm
wpfwpppm
CCCCC
CCCCC
???
??? (4-20)
F,J,Wasp
??
?
?
?
?
??
?
?
?
?
???
???
?
)(1 0 02
)(
1 0 0
2
2
pf
v
pf
pf
v
pf
fm C
C
????
????
??
(4-21)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
6,Heat and Mass Transfer
Ranz- Marshell:
33.05.0
33.05.0
Re6.02
PrRe6.02
ScSh
Nu
p
p
??
?? (4-22)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
§ 2,Forces Acted on Particles
1,Friction
? ?ppdpr vvvvCrF ??? ?? 22 (4-23)
Stokes,Cd = (1 + Rep2/3).24/Rep (1<Rep<1000) (4-24)
Newton Cd = 0.44 (Rep > 1000) (4-25)
where Rep = v - vp dp / n
when Tp > Tf,n = (1/3)np +(2/3)nf (4-26)
when mass of particles change,
Cd = Cd0ln(1 + B)/B
ln(1 + B) = m / (?dpNuD?f) (4-27)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2,Gravity Force and Buoyancy
Gravity,Fg = (1/6)?dp3?pg (4-28)
Buyancy,Fa = (1/6) ?dp3?fg (4-29)
From force equilibrium,
Fa + Fr
Fg
F
246
1
6
1 2233 pgp
dfppp
p
p
udCgdgd
dt
dum ?????? ????
Terminal subsidence velocity of particle
df
fpp
t
p
C
d
u
dt
du
?
??
3
)(4
,0
?
?? (4-30)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
up = 0,ut = uf
ut > uf,particle settle
ut < uf,particle rise
From (4-30)
f
fp
f
p
df
pt gd
C
du
?
??
nn
????
2
3
2
22
3
4
define
Ar
C
gd
Ar
du
d
t
f
fp
f
ppt
t
3
4
Re
,Re
2
2
3
?
?
???
?
??
nn
(4-31)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
When Rep <<1,
ArC t
p
d 18
1Re,
Re
24 ??
In general,Togec:
5.061.018Re Ar
Ar
t ??
(4-32)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
3,Force of Pressure Gradient
? ?
?
??
?
?
????
?
?
???
?
?
?
0
3
0
3
4
)c o s1(
x
p
V
x
p
rdspF
x
p
rpp
ppp
p
(4-33)
y
x
Fp
dp/dx
0
p0
4,False mass force
when particles accelerate relative to fluid not only
the velocity of particles faster,but also the velocity of
fluid around particles increased.
So F > mpap,it seams as the mass of particles
increased,
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
?
?
?
?
?
?
?
?
???
dt
vd
dt
vd
VF pfpfvm ?
2
1 (4-34)
In fact,
?
?
?
?
?
?
?
?
??
?
dt
vd
dt
vd
VkF
FF
pf
pfmvm
t h o rvmr e a lvm
?
,,,
(4-35)
Odar
? ?
pp
pf
c
c
m
da
vv
A
A
k
?
?
?
??
12.0
066.0
05.1
2
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
5,Basset Force
When particles move in static viscous fluid in straight
line,they suffer also a transient fluid friction which
includes the acceleration procedure of particles,This
force is caused by viscosity and related with non-static
state of fluid flow.
Ordaer
? ?? ?
?
?
t
pf
fpB
B dt
d
dv
d
dv
dk
F ?
?
??m??
4
2
(4-36)
Experiential correlation
? ? 31
12.388.2
?
??
c
B Ak
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
6,Magnus lift when particle turning
w = 2? 1000/s
reasons,- velocity gradient in fluid flow
- non-regular flow of particles
- collision among particles or with walls
- turning induced by heat and mass transfer
1)friction movement 383
ppf rM w?m??
2) Turning lift
(Magnus effect) w?? ffpi vdF 3
3
1?
(4-37)
(4-38)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
7,Saffman lift
Saffman lift is induced by movement of particles in
fluid flow with velocity gradient,
? ? ? ? 2/122/161.1 ??
?
?
???
???
dy
duvvdF f
pfpfs f?m (4-39)
8,热泳力
颗粒处在有温度梯度的流场中，将受到自高温区
的热压力而向低温区迁移，这种现象称为热泳，使
颗粒由高温区向低温区运动的力通常称为热泳力。
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From experiment,
1,Fth is more obvious to small particles,
2,DT↑,Fth↓
Fth = f (grad T,dp,mf,?f,?p,l,…)
l - free path of movement of particles
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?? T
r
l
G
r
l
G
r
l
C
r
T
F
pp
f
pp
f
p
m
p
ff
f
th
22131
1
9
2
?
?
?
?
?
m
? (4-40)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
In boundary layer of combustion chamber,
8.6,5
1,15,11 0 0 0
,105
,1 0 0
??
???
????
??
l
th
p
l
th
p
gthp
gthp
F
F
md
F
F
md
s
FFmd
FFmd
m
mw
m
m
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
9,Static Electricity Force
Reason,1) collision
2) gas iron diffusion
3) contact with surface change electricity
Fe = q E (4-41)
Lagrangian Movement Equation
?? Fdtvdm pp
(4-42)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
§ 3,Basic Equations
1,Statistic Conversion Equation of Mass of Particles
In general,the number of particles at
dtxdddddTvdtxdTv
ddTTTddddvdvv
pppppppp
pppppppppppp
)(),,,,,(
,),(,
???
??? ????????
For Louville Principle in Statistic Mechanics,for
dilute suspension,
0)( ?
?
??
?
?
?
??
?
??
?
??
?
??
?
?
dt
xd
xt
T
Tdt
d
dt
dd
ddt
vd
vt
p
p
p
p
p
p
p
p
???
?
????
(4-43)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
In fact,this equation is very difficult to solve,the
continues size distribution is separated into limited
number of size group,in which the particles have
same properties.
2,Control volume
Condition dk << l << L
3,Concept of volume average
For general variable ?k,which can be scale,
vector or tensor
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
kk
k
dV
kk
dVdV
dVdV
Vd
dV
k
??
?
??
?
?
/
~~1~
??? (4-44)
(k=1,2,3,…,f)
Eq.
?
?
?
? ?
???
?
?
?
?
?
??
?
?
?
?
?
???
?
?
?
?
?
?????
k
k
k
k
k
dA
kjkj
j
kj
j
dA
kik
k
k
i
k
dA
kjsjk
k
k
dV
k
dV
kk
k
k
kkk
Adn
dVxx
Adn
dVxx
Adnv
dVtt
dV
dVdV
dV
dV
dV
~~1
~
~~
~~1
~
~~
~
~
~1
~
~
~1~1~
?
?
?
?
?
?
?
?
?
?????
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
vsj is displace velocity of interphase,
nki or nkj is outer vector on interphase
In Eular coordinate the volume averaged
conversation equations for laminar multiphase flow.
Fluid phase continues equation
? ? Svxt i
j
?????? ?? (4-48)
Conti-eq,For k particles
? ? kkjkk Sv
tt
?
?
??
?
? ??
(4-49)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Continue-eq,for combined mediu,
? ? ? ? 0??? ????? ?? kjkj
j
k vvxt ????
(4-50)
? ? ? ? ? ?
iMiiki
rk
k
i
j
ji
i
ij
j
i FSvvvgxx
pvv
xvt ? ??????
??
?
???
?
??
?
?
?
?????
Momentum eq,for fluid
(4-51)
Momentum eq,for k particles,
? ? ? ? ? ?
ii Mkkikii
rk
k
ikkikjk
j
kik FSvvvgvvxvt ??????
??
?
?
?
????
(4-52)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Momentum eq,for combined medium,
? ? ??? ??????????????? ?? ?????
k
iki
j
ji
ik
kikjkij
j
kiki ggxxvvvvxvvt ??
?????
(4-53)
Energy eq,for fluid
? ? ? ? TScQnqQwxTxTcvxTct p
k
kkrss
jj
pj
j
p ??????
?
?
???
?
?
?
?
??
?
??
?
? ????
(4-54)
Where Qk - convection between particles and gas
cpTS - source term induced by phase change
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Energy eq,for k particles,
? ? ? ? ? ? kkprkkhkkkkjk
j
kkk STcQQQnTcvxTct ?????
??
?
? ?? (4-55)
??
??
????
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
k
rkk
k
hkrss
jj
k
kkkjkpj
jk
kkkp
QnQnqQw
x
T
x
TcvTcv
x
TcTc
t
?
????
Energy eq,for combined medium,
(4-56)
Where Qh - heat induced during phase change
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Particle conversion eqs in Lagrangian coordinate,
? ?
? ?
kk
kkpkrkkh
k
k
k
k
kii
ki
mi
i
rk
kii
k
ki
A
kknk
cm
TcTcmQQQ
dt
dT
m
m
vv
mn
F
g
vv
dt
dv
c o n s tNdAvn
i
????
?
?
????
?
?
???
?
,
.
?
(4-57)
(4-58)
(4-59)
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Chapter 4 Gas - Solid Two-Phase Flow
Examples,pulverized coal
soot
dust
ash
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
§ 1 Characteristics of Gas-solid Two Phase Flow
Rosin-Pammler Formula
? ? ? ?? ?nkk dddR /e x p ?? (4-1)
R(dk) - weight percent of particle with d > dk
n - index of non-evenness
d - characteristic diameter
? ? ? ? ? ?? ?
n
k
n
kn
k
ddd
d
n
dd
dR /e x p
)(
)1( ?? ?(4-2)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Particle average dimension:
radii average d10 = Snkdk/Snk (4-3)
area average d20 = (Snkdk2/Snk)1/2 (4-4)
volume average d30 = (Snkdk3/Snk)1/3 (4-5)
Sauter average d32 = Snkdk3/(Snkdk2) (4-6)
Typical particle size:
coal in fluidized beds 0 - 10 mm
pulverized coal 1 - 100mm
soot 1 - 10 mm
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2,Density
?m - combined density,?m = Mm/Vm = (Mp + Mf)/Vm
?f - superficial density of fluid,?f = Mf /Vm
?p,?k - superficial density of particle,?p = Mp/ Vm
?p - real density of particle
relation,
? ?? ??????? pkkfkfpfm dn ???????? 6/3
(4-7)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
3,Concentration
Volume concentration
f
p
v
pf
p
v
V
V
C
VV
V
C
?
?
?
(4-8)
Mass concentration
f
p
w
pf
p
w
M
M
C
MM
M
C
?
?
?
(4-9)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
In general,
wv
pm
CC ??
? ???
In pulverized coal transportation
38.0,26.0
1
10,5.0
m i n
3
??
???
??
?
r a n
v
m
f
vw
C
V
V
CC
??
?
(4-10)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
4,Viscosity
A,Einstein:
2)1(
5.21
v
v
f
m
C
C
?
??
m
m (4-11)
When Cv is not high
vfm C5.21 ??m
m
Cv = 10 -20%,F,Guth
21.145.21
vv
f
m CC ???mm
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Cv >20%,D,G,Thomas,
vCvv
f
m eCC 6.162 0275.005.105.21 ????mm
5,Specific heat and conductivity
)1(
)1(
wvfwvpcm
wpfwpppm
CCCCC
CCCCC
???
??? (4-20)
F,J,Wasp
??
?
?
?
?
??
?
?
?
?
???
???
?
)(1 0 02
)(
1 0 0
2
2
pf
v
pf
pf
v
pf
fm C
C
????
????
??
(4-21)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
6,Heat and Mass Transfer
Ranz- Marshell:
33.05.0
33.05.0
Re6.02
PrRe6.02
ScSh
Nu
p
p
??
?? (4-22)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
§ 2,Forces Acted on Particles
1,Friction
? ?ppdpr vvvvCrF ??? ?? 22 (4-23)
Stokes,Cd = (1 + Rep2/3).24/Rep (1<Rep<1000) (4-24)
Newton Cd = 0.44 (Rep > 1000) (4-25)
where Rep = v - vp dp / n
when Tp > Tf,n = (1/3)np +(2/3)nf (4-26)
when mass of particles change,
Cd = Cd0ln(1 + B)/B
ln(1 + B) = m / (?dpNuD?f) (4-27)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2,Gravity Force and Buoyancy
Gravity,Fg = (1/6)?dp3?pg (4-28)
Buyancy,Fa = (1/6) ?dp3?fg (4-29)
From force equilibrium,
Fa + Fr
Fg
F
246
1
6
1 2233 pgp
dfppp
p
p
udCgdgd
dt
dum ?????? ????
Terminal subsidence velocity of particle
df
fpp
t
p
C
d
u
dt
du
?
??
3
)(4
,0
?
?? (4-30)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
up = 0,ut = uf
ut > uf,particle settle
ut < uf,particle rise
From (4-30)
f
fp
f
p
df
pt gd
C
du
?
??
nn
????
2
3
2
22
3
4
define
Ar
C
gd
Ar
du
d
t
f
fp
f
ppt
t
3
4
Re
,Re
2
2
3
?
?
???
?
??
nn
(4-31)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
When Rep <<1,
ArC t
p
d 18
1Re,
Re
24 ??
In general,Togec:
5.061.018Re Ar
Ar
t ??
(4-32)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
3,Force of Pressure Gradient
? ?
?
??
?
?
????
?
?
???
?
?
?
0
3
0
3
4
)c o s1(
x
p
V
x
p
rdspF
x
p
rpp
ppp
p
(4-33)
y
x
Fp
dp/dx
0
p0
4,False mass force
when particles accelerate relative to fluid not only
the velocity of particles faster,but also the velocity of
fluid around particles increased.
So F > mpap,it seams as the mass of particles
increased,
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
?
?
?
?
?
?
?
?
???
dt
vd
dt
vd
VF pfpfvm ?
2
1 (4-34)
In fact,
?
?
?
?
?
?
?
?
??
?
dt
vd
dt
vd
VkF
FF
pf
pfmvm
t h o rvmr e a lvm
?
,,,
(4-35)
Odar
? ?
pp
pf
c
c
m
da
vv
A
A
k
?
?
?
??
12.0
066.0
05.1
2
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
5,Basset Force
When particles move in static viscous fluid in straight
line,they suffer also a transient fluid friction which
includes the acceleration procedure of particles,This
force is caused by viscosity and related with non-static
state of fluid flow.
Ordaer
? ?? ?
?
?
t
pf
fpB
B dt
d
dv
d
dv
dk
F ?
?
??m??
4
2
(4-36)
Experiential correlation
? ? 31
12.388.2
?
??
c
B Ak
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
6,Magnus lift when particle turning
w = 2? 1000/s
reasons,- velocity gradient in fluid flow
- non-regular flow of particles
- collision among particles or with walls
- turning induced by heat and mass transfer
1)friction movement 383
ppf rM w?m??
2) Turning lift
(Magnus effect) w?? ffpi vdF 3
3
1?
(4-37)
(4-38)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
7,Saffman lift
Saffman lift is induced by movement of particles in
fluid flow with velocity gradient,
? ? ? ? 2/122/161.1 ??
?
?
???
???
dy
duvvdF f
pfpfs f?m (4-39)
8,热泳力
颗粒处在有温度梯度的流场中，将受到自高温区
的热压力而向低温区迁移，这种现象称为热泳，使
颗粒由高温区向低温区运动的力通常称为热泳力。
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From experiment,
1,Fth is more obvious to small particles,
2,DT↑,Fth↓
Fth = f (grad T,dp,mf,?f,?p,l,…)
l - free path of movement of particles
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?? T
r
l
G
r
l
G
r
l
C
r
T
F
pp
f
pp
f
p
m
p
ff
f
th
22131
1
9
2
?
?
?
?
?
m
? (4-40)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
In boundary layer of combustion chamber,
8.6,5
1,15,11 0 0 0
,105
,1 0 0
??
???
????
??
l
th
p
l
th
p
gthp
gthp
F
F
md
F
F
md
s
FFmd
FFmd
m
mw
m
m
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
9,Static Electricity Force
Reason,1) collision
2) gas iron diffusion
3) contact with surface change electricity
Fe = q E (4-41)
Lagrangian Movement Equation
?? Fdtvdm pp
(4-42)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
§ 3,Basic Equations
1,Statistic Conversion Equation of Mass of Particles
In general,the number of particles at
dtxdddddTvdtxdTv
ddTTTddddvdvv
pppppppp
pppppppppppp
)(),,,,,(
,),(,
???
??? ????????
For Louville Principle in Statistic Mechanics,for
dilute suspension,
0)( ?
?
??
?
?
?
??
?
??
?
??
?
??
?
?
dt
xd
xt
T
Tdt
d
dt
dd
ddt
vd
vt
p
p
p
p
p
p
p
p
???
?
????
(4-43)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
In fact,this equation is very difficult to solve,the
continues size distribution is separated into limited
number of size group,in which the particles have
same properties.
2,Control volume
Condition dk << l << L
3,Concept of volume average
For general variable ?k,which can be scale,
vector or tensor
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
kk
k
dV
kk
dVdV
dVdV
Vd
dV
k
??
?
??
?
?
/
~~1~
??? (4-44)
(k=1,2,3,…,f)
Eq.
?
?
?
? ?
???
?
?
?
?
?
??
?
?
?
?
?
???
?
?
?
?
?
?????
k
k
k
k
k
dA
kjkj
j
kj
j
dA
kik
k
k
i
k
dA
kjsjk
k
k
dV
k
dV
kk
k
k
kkk
Adn
dVxx
Adn
dVxx
Adnv
dVtt
dV
dVdV
dV
dV
dV
~~1
~
~~
~~1
~
~~
~
~
~1
~
~
~1~1~
?
?
?
?
?
?
?
?
?
?????
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
vsj is displace velocity of interphase,
nki or nkj is outer vector on interphase
In Eular coordinate the volume averaged
conversation equations for laminar multiphase flow.
Fluid phase continues equation
? ? Svxt i
j
?????? ?? (4-48)
Conti-eq,For k particles
? ? kkjkk Sv
tt
?
?
??
?
? ??
(4-49)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Continue-eq,for combined mediu,
? ? ? ? 0??? ????? ?? kjkj
j
k vvxt ????
(4-50)
? ? ? ? ? ?
iMiiki
rk
k
i
j
ji
i
ij
j
i FSvvvgxx
pvv
xvt ? ??????
??
?
???
?
??
?
?
?
?????
Momentum eq,for fluid
(4-51)
Momentum eq,for k particles,
? ? ? ? ? ?
ii Mkkikii
rk
k
ikkikjk
j
kik FSvvvgvvxvt ??????
??
?
?
?
????
(4-52)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Momentum eq,for combined medium,
? ? ??? ??????????????? ?? ?????
k
iki
j
ji
ik
kikjkij
j
kiki ggxxvvvvxvvt ??
?????
(4-53)
Energy eq,for fluid
? ? ? ? TScQnqQwxTxTcvxTct p
k
kkrss
jj
pj
j
p ??????
?
?
???
?
?
?
?
??
?
??
?
? ????
(4-54)
Where Qk - convection between particles and gas
cpTS - source term induced by phase change
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Energy eq,for k particles,
? ? ? ? ? ? kkprkkhkkkkjk
j
kkk STcQQQnTcvxTct ?????
??
?
? ?? (4-55)
??
??
????
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
k
rkk
k
hkrss
jj
k
kkkjkpj
jk
kkkp
QnQnqQw
x
T
x
TcvTcv
x
TcTc
t
?
????
Energy eq,for combined medium,
(4-56)
Where Qh - heat induced during phase change
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Particle conversion eqs in Lagrangian coordinate,
? ?
? ?
kk
kkpkrkkh
k
k
k
k
kii
ki
mi
i
rk
kii
k
ki
A
kknk
cm
TcTcmQQQ
dt
dT
m
m
vv
mn
F
g
vv
dt
dv
c o n s tNdAvn
i
????
?
?
????
?
?
???
?
,
.
?
(4-57)
(4-58)
(4-59)
