A stochastic model of bubble distribution in gas-solid fluidized beds

Metallurgy

Journal of University of Science and Technology Beding Volume 13, Number 3, June 2006, Page 222

A stochastic model of bubble distribution in gas-solid fluidized beds Yanping Zhang’,2)and Li Wang” 1) Mechanical Engineering School, University of Science and Technology Beijing, Beijing 100083, China 2) Shougang Research Institute of Technology, Beijing 100043, China (Received 2005-03-03)

Abstract: On the basis of the Langevin equation and the Fokker-Planck equation, a stochastic model of bubble distribution in a gas-solid fluidized bed was developed. A fluidized bed with a cross section of 0.3 mx0.02 m and a height of 0.8 m was used to investigate the bubble distribution with the photographic method. Two distributors were used with orifice diameters of 3 and 6 mm and opening ratios of 6.4% and 6.8%,respectively. The particles were color glass beads with diameters of 0.3,0.5 and 0.8 mm (Geldart group B particles). The model predictions are reasonable in accordance with the experiment data. The research results indicated that the distribution of bubble concentration was affected by the particle diameter, the fluidizing velocity, and the distributor style. The fluctuation extension of the distribution of bubble concentration narrowed as the particle diameter, fluidizing velocity and opening ratio of the distributor increased. For a given distributor and given particles the distribution was relatively steady along the bed height as the fluidizing velocity changed. Key words: fluidization; bubble; stochastic force; bubble distribution

1. Introduction The behavior of bubbles in a gas-solid fluidized bed has typical random characteristics [ 1-31. In recent years, many researchers have further studied such phenomena of bubbles, with chaotic characteristics and dissipative structures [4- 121, and have analyzed the stochastic forces in fluidized beds [13-141. Research on the ru!es of movement and distribution of bubbles caused by stochastic force in a fluidized bed will help to understand the mechanism of fluidization on stochastic level.

2. Stochastic model of the distribution of bubble concentration in gas-solid fluidized beds As a result of the effect of interior and exterior stochastic complications the bubble distribution in a fluidized bed is instable and inhomogeneous. The instability is caused by the bubble diffusion and bubble excursion, and it is inhomogeneous because of the bubble coalescence and bubble abruption. The diffusion and excursion of bubbles always go with their coalescence and abruption, and this is the cause of the fluctuation of bubble distribution. Suppose the distributing of bubbles in a fluidized bed was a Markov process. On the basis of the Markov Corresponding author: Yanping Zhang, E-mail: [email protected]

process and the Langevin equation, the distribution of bubble concentration along the axis of a fluidized bed can be described by the Fokker-Planck equation:

ac a 2 c -=Dr--ubat

ac aZ

az2

where c is the distribution of bubble concentration in a fluidized bed, c=

. Hf

P [pdz

where p is the bubble concentration, Hf is the expanded bed height; D, is the axial diffusion coefficient; and ub is the bubble rise velocity. Considering the bubble generation, growth and abruption and the coalescence of two bubbles, Eq. (1) can be modified as:

where qo = Fro / Ipdz , Fro is the bubble generation frequency on the distributor; D, fusion and

ub-

ac az

a2c

az

is the bubble dif-

is the bubble convection; / l h c is

bubble attenuation. The coefficient of bubble attenua-

Y.P. Zhung et d,A stochastic model of bubble distribution in gasaolid fluidized beds tion is expressed by the following equation, p b

(3)

= kbFr0

where kh stands for the effect of bubble coalescence and a break of the bubble number.

On the basis of the two-phase theory the gas superficial velocity u, in the emulsion phase equals the minimum fluidizing velocity umf. Therefore NbVbUb = u-l(mf (1-NbVb)

(4)

where Nb is the number of the bubbles per unit volume; and v b is the volume of a bubble. The whole volume of bubbles in a fluidized bed can be obtained by,

H f- H o =

6‘‘

NbVbdH

(5)

conditions, the model of the distribution of bubble concentration is constructed.

3. Experimental The behavior of bubbles in a two-dimensional gassolid fluidized bed was investigated by the photographic method. The fluidized bed is a rectangular, cross-section bed, made in plexiglass, with an inner cross-section of 0.3 mx0.02 m and a height of 0.8 m. Solid particles are color glass with diameters of 0.3,0.5, and 0.8 mm (Geldart group B particles). Two distributors were used with orifice diameters of 6 mm (for type A distributor) and 3 mm (for type B distributor) and opening ratios 6.8% (for type A distributor) and 6.4% (for type B distributor), respectively. Fig. 1 is the schematic of the experimental equipment. The digital video camera was set in front of the fluidized bed, and the spotlight was set at the back of the bed. The video camera recorded the bubbles’ formation, motion, merging, and abruption.

where H o is the initial bed height. From Eqs. (4) and (9,we can get x

223

u-umf

kbPo -Db3 = +C 6 u b -umf ~

In addition, supplementary equations are (7)

A processing software for photographic data was developed. Its function was dispersing the consecutive images, determining bubbles’ coordinates, and statistically comparing experimental data.

(9) where DbOand Db are the initial bubble diameter and bubble diameter, respectively; uw is the initial bubble velocity; and K d . 7 1 1. Fig. 1. Schematic of the experimental equipment. l-compressor; Mransducer; 3-pressure measuring vessel; 4-spotlight and lens hood; M d i c o n ; bcomputer; 7--enemoscope; S-bellows; 9-fluidd bed; lo-grid.

(1 1)

where A is the area of the distributor and No is the number of orifices on the distributor. The boundary and initial conditions are

dC

jj

i=zg

= 90

u

> Unlf

(13)

where is the initial bubble generating position from above the distributor. By Eqs. (2), and (6)-(9), the boundary, and initial

4. Comparison of model predictions with experimental data Figs. 2-4 compare the model predictions with the experimental data. The model can basically describe the trend of the bubble concentration distribution. Fig. 2 shows the distribution of bubble concentration along bed height for diversified experimental conditions with type A distributor. It can be seen from the figures that the particle diameter, the fluidizing velocity and the distributor style affect the distribution of bubble concentration, and the bubbles distribute inhomogeneously along bed height. With the particle diameter increasing at the peak of the bubble concen-

J. Univ. Sci. Technol. Beging, VoL13, No.3, Jun 2006

224

tration the distribution lowers because the entire number and volume of bubbles decrease for the same fluidizing velocity. Therefore the probability of the coalescence and abruptions reduce. I4

1

(a)

1-i

- Model prediction Expenment data with type A distributor

1L

I

50

(a)

-Model prediction

40 -

Experiment data with type A distributor

30 u

20 10 -

0.10

0.06

0.14

. * .

.

0.18

I

0.b *'

O'

hlm

I

1.25 1.75 u l (m.s-l)

2.25

Expenment data w t h type A distnbutor Expenment data with type A distributor

61

41k., 101

2

..,

0.06

91"'

0.10

0.14

., .

~

fr 0

0.18

hlm

,

,

0.75

,

.,

1.75

1.25

_1

,

2.25

u l (rn.s-')

.

- Model

prediction Expenment data with type A distnbutor Expenment data with type A distributor

15

-

t 5-

0.05

;

0.10

0.15

. . .

0.20

I

I

I

I

I

.

I

hlm

Fig. 2. Trend of the bubble concentration along bed height: (a) u=1.7 d s , d,=0.3 mm; (b) ~ 1 . 7 5d s , d,=O.5 mm; (c) u=1.7 d s , d,=0.8 mm.

Fig. 3 shows the distribution of bubble concentration along with fluidizing velocity at some certain heights from above the type A distributor. The distribution of bubble concentration is basically steady with the changes of fluidizing velocity. With the fluidizing velocity increase the numbers of bubbles in different parts of the bed and the number of bubbles in the entire bed increase so the change of the distribution of bubble concentration is smooth. On the other hand, as the bubble numbers increase the coalescence enhances so the increase of the distribution of bubble concentration is restrained. Fig. 4 compares the distributions of bubble concentrations between two kinds of distributors. The bubble concentration peak with the type B distributor is lower than that with the type A distributor. Because the orifi-

2o

(d) -Model

15 0

prediction Expenment data with type A distributor

10-

5I

I

.I

I

I I

Fig. 3. Trend of the bubble concentration along with fluidizing velocity: (a) h=60 mm, d,=0.03 mm; (b) h=80 mm, d,=0.03 mm; (c) h=60 mm, d,=0.05 mm; (d) h=lW mm, d,=O.O5 mm.

5. Conclusions (1) A stochastic model for the distribution of bubble concentration in a gas-solid fluidized bed was proposed.

Y.P. Zhung et aL, A stochastic model of bubble distribution in gas-solid fluidized beds

The model predictions showed good agreement with experimental data.

-

15

- -Type B distributor

,b--\

,

0.10

0.06

0.14

0.18

hlm ~

12

- -

:I& 0.06

c--

Type A distributor -Type B dismbutor

,I

- - - - - - - -, - _, _ _ _ ,

0.10

0.14 hlm

0.18

Fig. 4. Comparing the bubble concentration between two kinds of grids: (a) dp=0.03mm, u=1.96 mls; (b) d,=O.OS mm, u=2.39 m/s.

(2) The distribution of bubble concentration is inhomogeneous along bed height and the fluctuation extension of the distribution of bubble concentration narrows as the particle diameter, fluidizing velocity, and opening ratio of the distributor increase.

(3) For a given distributor and given particles the distribution is relatively steady along bed height as the fluidizing velocity changes.

References [I] C.M. Van den Bleek and J.C. Schouten, Can deterministic chaos create order in fluidized bed scale up? Chem. Eng. Sci., 48( l993), No. 13, p.2367.

225

[2] N. Devanathan and A. Lapin, Chaotic flow in bubble column reactors, Chem. Eng. Sci.,50(1995), No.16, p.2661. [3] H.P. Cui, J.H. Li, H.Z. An, etal., Random characteristics of the heterogeneous structure in gas-solid fluidization, Sci. China Ser. B, 41(1998), No.4, p.377. [4] J.C. Schouten, M.L.M. van der Stappen, and C.M. van den Bleek, Scale-up of chaotic fluidized bed hydrodynamics, Chem. Eng. Sci., 51( 1996), p.1991. [5] C.M. van den Bleek and J.C. Schouten, Deterministic chaos: a new tool in fluidized bed design and operation, Chem. Eng., 53( 1993), p.75. [6] D. Bai, H.T. Bi, and J.R. Grace, Chaotic behavior of fluidized beds based on pressure and voidage fluctuations, AIChE J., 43(1997), p.1357. [7] D.V. Pence, D.E. Beasley, and J.B. Riester, Deterministic chaotic behavior of heat transfer in gas fluidized beds, J. Heat Transfer, 1l7( 1993, p.465. [8] J.H. Gao, X.G. Wang, and G. Hu. Control of spatiotemporal chaos by using random itinerant feedback injections, Phys. Lett. A, 283(2001), p.342. [9] D. Ohara, H. Ji, K. Kuramoto, et al., Chaotic characteristic of local voidage fluctuation in a circulating fluidized bed, Can. J. Chem. Eng., 77( 1999), p.247. [lo] S.S.E.H. Elnashaie, H.M. Harraz, and M.E. Abashar, Homoclinical chaos and the period-adding route to complex non-chaotic attractors in fluidized bed catalytic reactors, Chaos Solitons Fractals, 12(2001), p. 1761. [ 111 J. Li and G. Qian, Gas-solid fluidization-a typical dissipative structure, Chem. Eng. Sci., 51(1996), No.4, p.667. [12] M. Qi and L. Wang, Nonequilibrium thermodynamic and dissipative structure mechanism of fluidized beds, J. Univ. Sci. Technol. Beijing, 6( 1999), No.2, p.90. 131 Y.A. Buyevich and S.K. Kapbasov, Random fluctuations in a fluidized bed, Chem. Eng. Sci., 49(1994), No.8, p.1229. 141 G. Hu, Stochastic Forces and Nonlinear Systems, Shanghai Scientific and Technological Education Publishing House, Shanghai, 1994, p.34. 151 P. Wu, L. Wang, X. Feng, et al., Surface-particle-emulsion heat transfer model between fluidized bed and horizontal immersed tube, J. Univ. Sci. Technol. Beijing, 9(2002), No.2, p.99.