Numerical simulations based on three dimensional discrete element model (DEM) are conducted for the mono-disperse, binary and ternary system of particles in a fluidized bed. Fluid drag force acting on each particle depending on its size and relative velocity is assigned. An expression for the drag coefficient corresponding to Ergun’s correlation is developed and applied to the system of fluidized bed with particle size ratios of 1:1 for the mono-disperse system, 1:1.2, 1:1.4 and 1:2 for the binary system as well as 1:1.33:2 for the ternary system by keeping total volume and surface area of the particles constant. Results indicated that a reasonable estimation of modified drag force is achieved in the fluid cells. Total translational kinetic energy of particles is found to be increasing with the corresponding increase in the particle size ratio, emphasizing an enhanced momentum transfer between the particles with size distribution. Systems with wide size distribution indicated higher particle velocities around the bubble resulting in the faster bubble growth and its subsequent transition through the fluidized bed. Interesting yet promising nature of these results for the particle systems with size distribution reveals the important trends in determining mixing and segregation of particles in the fluidized bed.

1. A Computational Study of Fluidized
beds with Particle Size Distribution
N. Tagami and M. Horio
Tokyo University of Agriculture and Technology
Department of Chemical Engineering
Tokyo, Japan
Presented at:
The Second Asian Particle Technology Symposium (APT 2003)
17th-19th December 2003, Penang, Malaysia
N.Tagami and M.Horio 19th/12/2003 1

2. Contents
1. Introduction
2. Modifications of fluid drag calculation
3. Calculation results
4. Conclusions
N.Tagami and M.Horio 19th/12/2003 2

3. Introduction
With our code SAFIRE, we have
demonstrated that the discrete element
method (DEM) can be a powerful tool for
industrial chemical reactor design issues.
However, so far, most of the work in the literature
has limited within uniformly sized particles.
There is insufficient consideration about the
effect of particle size distribution (PSD)
present in a fluidized bed
N.Tagami and M.Horio 19th/12/2003 3

4. What happens with the introduction of
thickness
PSD ?
(1) non-even (1) Fluid drag acting on each
fluid drag particle should be assigned
depending on relative velocity
(2) 2D → 3D and particle size.
(2) Three dimensional calculation
(3) fluid drag becomes inevitable
dependency 2D → 3D (3) Drag force is assigned to each
on alignment particle depending on the
particle alignment
In this work SAFIRE was
modified in terms of (1) and (2).
N.Tagami and M.Horio 19th/12/2003 4

5. Determination of CD from fixed bed data
Pressure drop in a dense phase is given by
Ergun(1952)
ΔP*  ΔP  ρ f gL A p : Projected area
d p : Particle diameter

1 - ε  150 1  ε μ f 
 1.75ρ f u  v  u  v  ε : Void fraction

d p  d p  ρ f : Fluid density
Equation of fluid motion for 1D steady flow:
ε
ΔP
 nFpf  ερ f g  0 n 1  ε / πd p3/6 
ΔL
Drag coefficient defined with mean diameter:
8 Fpf 2001  ε μ f
CD  C D,Ergun   2.33
 d p 2ρ f u  v
2 d pρ f ε u  v
N.Tagami and M.Horio 19th/12/2003 5

6. Approximate expression for CD
corresponding to Ergun correlation
extension for individual particle
2001  ε μ f 2001  ε μ f
C D,Ergun   2.33 C D,Ergun   2.33
d pρ f ε u  v d pρ f ε u  v
extension to a system with a wide PSD
200μ f 1  ε 
CD,Ergun   2.33
d pρ f u  v ε
effect of  could be different in the mixed particle
system, but let’s use the same expression
N.Tagami and M.Horio 19th/12/2003 6

7. Drag coefficients
Apparent drag coefficient [-]
10000
Dense phase From Wen-Yu Eq.
2001  ε μ f 1000
C   2.33 Single particle
d pρ f ε u  v
D, Ergun
100
Dilute phase 10
Wen-Yu(1966) correlation 1
0.4
CD, WY  ε 3.7CD,s 0.6
Vo
0.8 From Ergun Eq. 1.2 1.6
id
0.8
ag
0.4
1.0 0.0
locity [m/s]
e[
where Interst itial fluid ve
-]
C D,s 
24
Re
 
1  0.15Re 0.687 Re  700
 0.44 Re  700
N.Tagami and M.Horio 19th/12/2003 7

8. Governing equations
Translational motion of particle
dv
m   Fcollision,pp   Fcollision,pw   Fcohesion  Ffp  mg
dt
Rotational motion of particle
dω
I   M collision,pp   M collision,pw   M cohesion  M fp   M wall
dt
F : Force
Equation of continuity for fluid I : Moment of inertia
ε εu  M : Moment
 0
t t m : Mass of a particle
u : Velocity of fluid
Equation of motion for fluid v : Velocity of a particle
 u u  σ  : Void fraction
ρf ε   u   ε  nFpf  ερ f g
 t x  x  : Stress tensor
 : Angular velocity
N.Tagami and M.Horio 19th/12/2003 8

9. Objectives of the present
computation
To
•confirm the present fluid-particle interaction
treatment satisfy Ergun correlation
macroscopically for systems with PSD.
•analyze the effect of PSD on macroscopic
fluidized bed behavior for cases with the same
mean particle size (dpsv) and total bed volume.
N.Tagami and M.Horio 19th/12/2003 9

10. Computational Conditions
dp1/dp2 [mm/mm] Number of particles
1.00 30000
1.10 / 0.917 (1.20) 11270 / 19474
1.20 / 0.857 (1.40) 8681 / 23819
1.50 / 0.750 (2.00) 4444 / 35556
The average surface to The total volume and surface
volume diameter is identical area of the particles are also
for each calculation as held constant
 N d   1.00 [mm] Vtotal  1.57 105 [m3 ]
3
dpsv= N d  p
 Stotal  9.43 10 2 [m 2 ]
2
p
N.Tagami and M.Horio 19th/12/2003 (continued)10

11. (continued) Linear Spring
Spring constant : 800N/m
Linear dashpot
Restitution coefficient : 0.9
Particle density : 2650 kg/m3
Friction coefficient : 0.3
50mm
Superficial velocity [m/s]
10mm 1.122
200mm
0.5
App. 54mm
0 1.0
Air
N.Tagami and M.Horio 19th/12/2003
Time[s] 11

12. Calculation results
1.10mm 11270 1.20mm 8681 1.50mm 4444
1.00mm 30000 0.917mm 19474 0.857mm 23819
0.750mm 35556
dp1/dp2= 1.2 1.4 2.0
N.Tagami and M.Horio 19th/12/2003 12

13. Comparison of fluid drag force
acting on each fluid cell
Blue zone: fluid drag force numerically determined
agrees with Ergun correlation + 20% in each fluid cell
-
(Fdrag coefficitent) / (FErgun,fluid cell)
dp1/dp2= 1.2 1.4 2.0
N.Tagami and M.Horio 19th/12/2003 13

14. Total translational kinetic energy [mJ] Total translational kinetic energy
0.8
0.6
Uniform system
Binary system
0.4
dp1/dp2=2.0
dp1/dp2=1.2
0.2
dp1/dp2=1.4
0.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Time [s]
Total translational kinetic energy increases
as the difference in particle size increases
N.Tagami and M.Horio 19th/12/2003 14

15. Cumulative number of collisions [#] Cumulative number of collisions
30000
25000 Uniform system
20000 dp=1.00mm
15000 Ten particles are
10000
5000
traced in each
0 component
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
30000 30000
25000 Binary system 25000 Binary system
20000 dp=1.10mm 20000 dp=0.917mm
15000 15000 dp1/dp2
10000 10000 = 1.2
5000 5000
0 0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
N.Tagami and M.Horio Time [s] 19th/12/2003 (continued) 15

16. (continued)
30000 30000
Cumulative number of collision [#]
25000 Uniform system
Binary system 25000 Binary system
In the binary system
20000 dp=1.00mm
dp=1.20mm 20000 dp=0.857mm
momentum transportation
15000 15000 dp1/dp2
10000 10000
between particles= 1.4
is
5000 5000 emphasized
0 0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
30000
30000
Binary system 25000 Binary system
25000
20000 dp=1.50mm 20000 dp=0.750mm
15000 15000 dp1/dp2
10000 10000 = 2.0
5000 5000
0 0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Time [s]
N.Tagami and M.Horio 19th/12/2003 16

17. Conclusions
To achieve the DEM simulation with PSD,
modification of fluid drag force calculation is needed.
In the present study, the drag force is
computed using the drag coefficient combined
with Ergun correlation.
The calculation results show that reasonable fluid
drag force is calculated for each fluid cell.
The bed motion activity increases due to
the existence of particle size distribution
N.Tagami and M.Horio 19th/12/2003 17