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Prediction of Core-Annulus Solids Mass Transfer Coefficient
in Gas-Solid Fluidized Bed Risers*
论文题目要精炼、醒目，去掉“Study on”的字样。
LIU Xinhua (刘新华)1,2, GAO Shiqiu (高士秋)1, ** and SONG Wenli (宋文立)1
1
Multiphase Reaction Laboratory, Institute of Process Engineering, CAS, Beijing 100080, China
2
Graduate school of the Chinese Academy of Sciences, Beijing 100864, China
Abstract Based on analysis of energy dissipation in the core region of gas-solid fluidized bed risers, a
simplified model for determination of core-annulus solids mass transfer coefficient was developed
according to turbulent diffusion mechanism of particles. The simulation results are consistent with
published experimental data. Core-annulus solids mass transfer coefficient decreases with increasing
particle size, particle density and solids circulation rate, but generally increases with increasing
superficial gas velocity and riser diameter. In the upper dilute region of gas-solid fluidized bed risers,
core-annulus solids mass transfer coefficient was found to change little with the axial coordinate in
the bed.
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Keywords solids mass transfer coefficient, core-annulus structure, turbulent diffusion, gas-solid
fluidization
Received 2004-12-14, accepted 2005-06-14.
* Supported by the National Natural Science Foundation of China ( 90210034, 20221603). 国家自
然科学基金等国家级资助项目应注明编号。基金英文名称书写完整正确。
** To whom correspondence should be addressed.
1
1 INTRODUCTION
Two main categories of hydrodynamic models of gas-solid fluidized bed risers
have been proposed. One predicts gas-solid flow structure from fundamental
conservation equations by using computational fluid dynamics. The other approximates
radial profile of the two phases by division of the flow into core and annulus.
Core-annulus models are widely used in analysis of radial non-uniform profiles of
gas-solid fluidized bed risers, but as a key model parameter for calculation of local
radial solids mass flux from the core to the annulus, core-annulus solids mass transfer
coefficient kca is difficult to be determined due to the complexity of involved
mechanisms and measuring methods. In previous literatures[1, 2], the deposition
coefficient of fine particles to the wall in the freeboard of bubbling fluidized beds kd
was usually taken as an approximation of kca and related to the amplitude of velocity
fluctuation u  :
kd 
0.1 πu 
1  S 12
(1)
公式依出现的顺序编号。物理量注意用斜体。
with the particle Stokes number S  pUg dp2 18 Dt  . Besides this, other relevant
researches include that Malcus and Pugsley[3] determined radial profiles of local
voidage and local solids mass flux simultaneously by using a sampling probe, and Liu
et al.[4, 5] did similar work by using phase Doppler particle analyzer (PDPA).
The mechanisms involved in radial mixing of particles in gas-solid fluidized bed
risers are different in different fluidization regimes. In the bubbling regime, radial
mixing of particles is mainly induced by particle convective movement[6]. In the
turbulent regime, turbulent diffusion due to eddies caused by rising bubbles may
explain radial solids mixing[7]. In the fast fluidization, radial mixing of particles is due
to particle fluctuation velocity resulting from gas-particle and/or particle-particle
interaction[8]. Although kca is believed to depend much upon turbulent fluctuation of
particles in dilute gas-solid suspensions[9, 10], a reasonable model to determine kca is
not available up to now.
2
In a gas-solid suspension, particle diffusion coefficient is closely related to
turbulent fluctuation and energy dissipation of the gas induced by gas-particle
interaction[11, 12]. Based on analysis of energy dissipation in the core region of
gas-solid fluidized bed risers, a mathematical model for prediction of kca is developed
according to turbulent transfer mechanism of particles in this study.
2 MODEL DEVELOPMENT
2.1 Mechanism analysis
As mentioned before, a core-annulus model [Fig.1(a)] （在正文中必须有与图、
表呼应的文字，且叙述应与图、表结果相符。图、表依出现的顺序编号）assumes
the flow in fluidized bed risers consists of a dilute up-flowing suspension of solids in
the center and a dense down-falling suspension of solids adjacent to the wall. This
model is able to approximate radial profiles of gas-solid fluidized bed risers, but it does
not consider particle clustering behavior. A two-phase model [Fig.1(b)] proposed by Li
et al.[13] which assumes the existence of particle clusters dispersed in a
homogeneously up-flowing dilute gas-solid mixture can predict well local flow
structure of particle-fluid two-phase flow. Bi[14] pointed out that a two-phase
two-region model seems to capture main flow features in gas-solid fluidized bed risers.
Many experimental results indicated that most so-called particle clusters in the core
region are only loose agglomerations of particles[15] and particle clusters exist only in
the annulus region. Therefore, this study assumes a simple two-phase two-region
model like that of Werther et al.[16], as shown in Fig. 1c.
(a)
(b)
(c)
Figure 1 Illustration of (a) a core-annulus model, (b) a two-phase model and (c) a simple
3
two-phase two-region model
When particles in the turbulent core region collide with particle clusters in the
laminar annulus region due to gas-particle and/or particle-particle interaction, they are
readily captured by the particle clusters due to the velocity difference between
them[17]. But it is not easy that some particles from the decomposition of unstable
particle clusters are re-entrained by the gas of high velocity in the core region. This
exchange mechanism of particles between the two regions seems to contribute
significantly to radial net transfer of particles from the core to the annulus despite an
adverse solids concentration gradient. Radial transfer of particles in gas-solid fluidized
bed risers is believed to be due to turbulent diffusion of particles in the core region[8—
10], not due to concentration difference diffusion.
2.2 Model formulation
Principle assumptions in this model are made as follows: (1) the gas-solid flow is
full developed; (2) the turbulence in the core region is homogeneous and isotropic; (3)
all the gas flows only through the core region; (4) the gas-solid slip velocity in the core
region is equal to the terminal velocity of a single particle.
2.2.1 Determination of turbulent fluctuation of the gas
Similar to the analysis of Li et al.[13], the total energy per unit mass of the gas in
the core region introduced by itself in unit time can be estimated according to
assumption (1) as follows:
ETc 

p
 g  1   c  gU c
g  c
(2)
For a dilute gas-solid two-phase flow in the core region, the energy for transporting and
suspending the solids is equal to
Estc 

p
 g  gGc
 p g  c
(3)
so the turbulent energy dissipation of the gas in the core region can be calculated by
4
Edc  ETc  Estc 

 g  g 
Gc 
1   c U c  
g  c
p 

p
(4)
By assumption (2), the energy dissipation of the gas in the core region can also be
expressed as[11]
Edc  C
34
μ
1.5u 
2 32
gc
(5)
Le
In order to account for the effects of higher solid concentration on the gas turbulence
and the drag coefficient respectively, Cao and Ahmadi[18] proposed that the coefficient
Cμ  0.09Cμ*
and
Cμ*  1  1   c   p

3
the
particle
 C  
3
m L
1
relaxation
time
 p   p* p dp2 18  ,
and  p*  1  0.1Re0.75
 1   c  Cm 1
p
1
2.5Cm
where
with the
particle Reynolds number Re p  g d p ut  for dilute gas-solid flow. The Lagrangian
integral time scale  L can be calculated from the turbulent integral length scale as
 L  Le ugc
[12].
Thus, fluctuation velocity of the gas in the core region u gc can be
obtained by combining Eqs. (4) to (5).
2.2.2 Deduction of turbulent diffusivity of the particles
Based on a Lagrangian flow analysis[11], turbulent diffusivity of the gas in the
core region can be defined as
 Le  ugc
2 L
Dgc  ugc
(6)
According to Hinze[11] and Tchen[19], the diffusivities of the gas and small discrete
particles in the core region can be related through their turbulent fluctuation:
Dpc
Dgc

2
upc
(7)
2
ugc
A less restrictive theory developed by Soo[20, 21] indicates that the ratio of fluctuation
velocity of small particles to that of the gas in a dilute gas-solid suspension can be
expressed as
2
upc
2
ugc
where
the
fluid-particle
 π 1 exp  2 erfc 1
interaction
parameter
  p dp2ugc  9L  .
(8)
The
5
complementary error function erfc   
the gas L can be calculated by

2
 e
π 
t 2
dt . The Lagrangian micro scale of
2
  L   12.5 / Re2λ  29 / Re λ , where the turbulent
Reynolds number based on the Eulerian micro scale Re λ   ugc g  and the Eulerian
micro scale   ugc 15  g Edc  [22].
0.5
Eqs. (6)－(8) are used to estimate the diffusivity of particles in the core region in
this study. Strictly speaking, the above analysis seems to be valid only for small
particles of not greater than the Kolmogorov micro scale, and this scale is about 100－
300 μm in gas-solid fluidized beds under ordinary operating conditions[23].
2.2.3 Calculation of core-annulus solids mass transfer coefficient
Based on the assumption of the turbulent Sherwood number of 4, Pemberton and
Davidson[1] obtained good calculation results in their studies, and they think that this
assumption is consistent with use of the 1/7 power law for the velocity profile in
turbulent flow. Similarly, this assumption is also taken for the gas-solid two-phase flow
in the core region in this study:
Shc 
kca Lc
4
Dpc
(9)
where Lc   Dt is the characteristic length of the core region. Eq. (9) is used to
estimate kca in gas-solid fluidized bed risers.
2.3 Estimate of model parameters
2.3.1 Superficial gas velocity in the core region
According to assumption (3), the superficial gas velocity in the core region Uc is
equal to
Uc 
Ug
(10)
2
where dimensionless core radius  can be determined by[24]
U D  
R
  c  1  1.1 g t g 
R
  
0.22
H
 
 Dt 
0.21
 H Z 


 H 
0.73
(11)
6
Despite neglecting the effect of solids circulation rate, the above correlation has been
confirmed by experimental data and computations[25].
2.3.2 Average axial solids mass flux in the core region
Since radial profile of local axial solids mass flux G can be approximated by[26]
  r m 
G
ma
 a 1      1 
Gs
ma  2
  R  
(12)
where m  5 and a   5  2 7  , average axial solids mass flux in the core region
1
Gc can be obtained by integrating the above equation between zero and Rc R ,
Gc  Gs 
1
π  Rc R 
2
Rc
R
0

G
5 5
r r
 2π   d   
Gs
5
Gs
 R   R  7  2
(13)
2.3.3 Average voidage in the core region
According to assumption (4), the gas-solid slip velocity is equal to the terminal
velocity of a single particle:
Uc
c

Gc
 ut
 p 1   c 
(14)
So, average voidage in the core region  c can be obtained by resolving Eq. (14) if Uc
and Gc are determined at first.
2.3.4 The turbulent integral length scale
In the core region of a gas-solid fluidized bed riser of diameter Dt, the turbulent
integral length scale may be estimated as Le  0.1 Dt for single phase flow[27, 28].
But the integral length scale seems to be proportional to turbulent fluctuation of
particles in gas-solid two-phase flow[29, 30]. It is generally considered that those
particles smaller than the Kolmogorov micro scale damp the turbulence, whereas those
particles greater than the Kolmogorov micro scale enhance the turbulence due to vortex
shedding[31, 32]. That is, the fluctuation velocity of small particles decreases with
increasing solids concentration. Therefore, the turbulent integral length scale Le is
assumed to be inversely proportional to solids concentration in this study,
Le  0.1 Dt
Cm  1   c 
Cm
(15)
7
where Cm is the maximum solids fraction for a random packing[33]. Eq. (15) shows
that Le is equal to 0.1 Dt for the core region at  c  1 , which is consistent with that
of single phase flow. If average solids concentration in the core region approaches Cm ,
the turbulent integral length scale Le would decrease to zero.
3 SIMULATION RESULTS
If the model parameters U c , Gc ,  c and Le are determined at first according
to Eqs. (10), (13), (14) and (15) respectively, fluctuation velocity of the gas in the core
region u gc can be obtained through trial-and-error according to Eqs. (4)－(5). Then kca
can be estimated from Eq. (9) after Dpc is calculated directly from Eqs. (6)－(8). All
the simulation calculations in this study were performed on the basis of the
experimental conditions of Arena et al.[34].
Core-annulus solids mass transfer coefficient kca generally increases with
increasing particle diffusivity and hence is proportional to the product of particle
fluctuation velocity u p and the mixing length lm [35],
kca  up lm
(16)
where lm is a function of average distance between two collisions of each particle. The
shorter the distance, the smaller the mixing length. This is called the mixing length
theory. Combined with this theory, the simulation results are qualitatively analyzed
here in detail.
As reported by previous literatures[36, 37], the influence of particle size d p on
kca acts in a similar way as particle density  p , and an increase of d p or  p leads to
decreasing u p due to weakening gas-particle interaction. Therefore, Core-annulus
solids mass transfer coefficient kca decreases obviously with increasing d p or  p
when lm changes little at fixed Ug and Gs[35], as shown in Fig. 2.
8
Figure 2 Variation of core-annulus solids mass transfer coefficient for the particles of different
density with particle size
(Ug=4.0m·s-1, Gs=96kg·m-2·s-1, ρg=1.1795kg·m-3, μ=1.89×10-5kg·m-1·s-1, H=11m, Dt=0.4m, Z=5.0m)
ρp, kg·m-3:
1750;
2540;
3000
（量、单位和符号严格执行国家标准，不可使用非法定计量单位。引用文献数据
出现非法定计量单位时，应加换算成法定计量单位的关系式。组合单位用指数形
式，如 J·kg-1，不用 J/kg 形式。数字与单位之间加空格）
坐标图一律采用封闭图，端线尽量取在刻度线上。
横、竖坐标必须垂直，坐标刻度线的疏密程度要相近，刻度线朝向图内，去掉无
数字对应的刻度线，不用背景网格线。标度数字尽量圆整，过大或过小时可用指
数表示，如 102、10-2。
When solids concentration increases, a decrease of average distance between two
collisions of each particle leads to decreasing lm, and u p for the particles smaller than
the Kolmogorov micro scale may also decrease. So a decreasing kca for the particles of
different size and density occurs due to decreasing  c when Gs increases at a constant
Ug, as shown in Fig. 3. On the contrary, although an increase of Ug on the one hand
leads to increasing  c and hence to increasing lm, and on the other hand a decreasing
trend of Uc and hence to decreasing u pc [38] due to increasing dimensionless core
radius  , core-annulus solids mass transfer coefficient kca gradually increases with
increasing Ug at a fixed Gs under the conditions in this study, as shown in Fig. 4.
9
Figure 3 Variation of core-annulus solids mass transfer coefficient for the particles of different
size and density with solids circulation rate
(Ug=4.0m·s-1, ρg=1.1795kg·m-3, μ=1.89×10-5kg·m-1·s-1, H=11m, Dt=0.4m, Z=5.0m)
dp=80μm, ρp=2540kg·m-3;
dp=120μm, ρp=2540kg·m-3;
dp=120μm, ρp=1750kg·m-3
Figure 4 Variation of core-annulus solids mass transfer coefficient for the particles of different
size and density with superficial gas velocity
(Gs=96kg·m-2·s-1, ρg=1.1795kg·m-3, μ=1.89×10-5kg·m-1·s-1, H=11m, Dt=0.4m, Z=5.0m)
dp=80μm, ρp=2540kg·m-3;
dp=120μm, ρp=2540kg·m-3;
dp=120μm, ρp=1750kg·m-3
According to Arena et al.[34], so-called core-annulus flow structure generally
forms at Z>3.0 m under the simulation conditions. Under the condition of forming
core-annulus structure, dimensionless core radius  increases with increasing Z,
which leads to decreasing Uc and increasing  c simultaneously. It is because the
above two contrary influences roughly counteract with each other that kca changes little
with Z at Z>3.0 m, as shown in Fig. 5. This phenomenon agrees well with previous
numerical simulations[39], indicating kca can be approximated as a constant in the
upper dilute region of gas-solid fluidized bed risers. While, an increase of Dt may lead
to increasing  [24, 40] and hence to increasing  c , so kca for the particles of different
size and density increases significantly with increasing Dt despite that u pc may
decrease with decreasing Uc caused by increasing dimensionless core radius  at
fixed operating conditions, as shown in Fig. 6.
10
Figure 5 Variation of core-annulus solids mass transfer coefficient for the particles of different
size and density with measuring height
(Ug=4.0m·s-1, Gs=96kg·m-2·s-1, ρg=1.1795kg·m-3, μ=1.89×10-5kg·m-1·s-1, H=11m, Dt=0.4m)
dp=80μm, ρp=2540kg·m-3;
dp=120μm, ρp=2540kg·m-3;
dp=120μm, ρp=1750kg·m-3
Figure 6 Variation of core-annulus solids mass transfer coefficient for the particles of different
size and density with the diameter of gas-solid fluidized bed risers
(Ug=4.0m·s-1, Gs=96kg·m-2·s-1, ρg=1.1795kg·m-3, μ=1.89×10-5kg·m-1·s-1, H=11m, Z=8.0m)
dp=80μm, ρp=2540kg·m-3;
dp=120μm, ρp=2540kg·m-3;
dp=120μm, ρp=1750kg·m-3
4 MODEL VALIDATION
Very few relevant experimental data are available because of the difficulties
involved in the measurement of kca besides that Bolton and Davidson[2] determined kd
as an approximation of kca in a gas-solid fluidized bed riser. In fact, core-annulus solids
mass transfer coefficient kca can also be deduced from εc and local radial solids mass
flux from the core to the annulus Gca according to its definition kca  Gca   p 1   c   .
Since Malcus and Pugsley[3] and Liu et al.[4, 5] determined radial profiles of local
radial solids mass flux and local voidage simultaneously under different operating
conditions respectively, local radial solids mass flux from the core to the annulus Gca
can be obtained directly from their experimental data according to experimental
determined or calculated core radius. Average solids concentration in the core region
11
can be calculated by integrating local solids concentration along radial direction
between the riser center and the core-annulus boundary. Accordingly, corresponding kca
can be calculated indirectly from their experimental data. For easy comparison with the
previous correlation, all the experimental and calculated results are plotted versus
superficial gas velocity, as shown in Fig. 7. Obviously, Eq. (1) fits the experimental
data of Bolton and Davidson[2] well, but appears to be invalid when operating
conditions change much, and the average and the maximum relative error of Eq. (1) are
even greater than 72% and 100%, respectively. On the contrary, the average and
maximum relative error of this model can be anticipated to be smaller than 24% and
40% respectively, and the model seems to be able to give better prediction over a wide
range of operating conditions because of taking account of the influences of much
more factors such as solids circulating rate, axial coordinate, riser diameter, etc.
Figure 7 Comparison of core-annulus solids mass transfer coefficient calculated from different
models with experimental data
Exp.: □ Ref. [4, 5]; ○ Ref. [2]; ∆ Ref. [3]
Cal.:
this model;
Eq. (1)
5 CONCLUSIONS
Under the condition of forming the core-annulus structure in gas-solid fluidized
bed risers, a simplified model for estimate of core-annulus solids mass transfer
coefficient was developed on the basis of lateral transfer mechanism of particles. The
prediction of this model is in good agreement with available experimental data and
gives many significant qualitative results over a wide range of operating conditions.
Introducing the turbulent integral length scale is an approximation for the dilute core
region of gas-solid fluidized bed risers, and many more experimental data are needed
12
to further verify this model.
NOMENCLATURE
按英文字母顺序排列，同一字母先排大写后排小写；希腊文接英文后排，也按字
母顺序排列。
符号与说明间用二字线，说明文字与单位间用逗号。
一个符号只代表一个物理含义，一个物理量只用一个符号表示。符号尽量简化，
最好以单字母表示。
物理量符号采用国家标准中的规定，如压力用 p、温度用 T，均用斜体。矢量、张
量、矩阵用黑斜体。
下角一般用小写正体，只有下列情况除外：
（1）表示数、变量用小写斜体，如 Si，
i=1,2,…，i 用斜体；
（2）保留原物理含义，如比定压热容 cp 中的 p 为小写斜体；
（3）液体 l 为区别数字 1，用斜体 l。
a
constant determined by dimensionless core radius
Cm
maximum solids fraction for a random packing (=0.64356)
D
radial diffusion coefficient, m2·s-1
Dt
diameter of the riser, m
dp
particle diameter, m
Ed
energy dissipation rate of the gas, J·kg-1·s-1
Est
energy for suspending and transporting the particles, J·kg-1·s-1
ET
total energy consumption of the gas, J·kg-1·s-1
G
local axial solids mass flux, kg·m-2·s-1
Gca
local radial solids mass flux from core to annulus, kg·m-2·s-1
Gs
solids circulation rate, kg·m-2·s-1
g
gravity acceleration, m·s-2
H
height of the riser, m
k ca
core-annulus solids mass transfer coefficient, m·s-1
kd
deposition coefficient of fine particles to the wall, m·s-1
lm
mixing length, m
Lc
characteristic length of the core region, m
Le
turbulent integral length scale, m
m
empirical constant (=5)
R
radius of the riser, m
S
Stokes number
Shc
mass transfer Sherwood number between core and annulus
U
superficial gas velocity, m·s-1
13
u
fluctuation velocity, m·s-1
ut
terminal velocity of a single particle, m·s-1
Z

measuring height, m

average voidage

gas viscosity, kg·m-1·s-1

density, kg·m-3
L
Lagrangian integral time scale, s
p
particle relaxation time, s
dimensionless core radius
Subscripts
c
core
g
gas
p
particle
REFERENCES
参考文献以在正文中引用的先后顺序排列。内部资料和非出版物不能引用。
参考文献全部采用英文著录，文献作者应全部列出，具体格式参见“文稿体例要
求”
。参考文献数量最好不少于 20 篇。
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