Mixing of randomly moving particles in liquid—solid fluidized beds

Powder Technology, 42 (1985) 145 - 152

Mixing o f R a n d o m l y Moving Particles

145

in Liquid-Solid Fluidized B e d s

N. YUTANI* and L. T. FAN Department of Chemical Engineering, Kansas State University, Manhattan, KS 66506 (U.S.A.) (Received May 26, 1983;in revised form August 2, 1984)

SUMMARY

The deterministic diffusion equation o f the Fickian type has been widely employed in modeling solids mixing in fluidized beds. In this work, a stochactic or statistical model has been proposed for the solids mixing in a liquid-solid fluidized bed o f relatively large particles. Specifically, the particle diffusivity under the steady fluidized state has been derived by combining the well-known 'Ornstein-Uhlenbeck' equation o f stochastic processes and the so-called partition function of particle velocities. The values o f particle diffusivity calculated from the model have been shown to agree well with the available experimental data. Furthermore, it has been shown that the solids-mixing process in the bed can be related to the time-dependent nature o f particle motion.

INTRODUCTION The deterministic diffusion equation of the Fickian type was widely employed for estimating the mixing coefficient or particle diffusivity in liquid-solid fluidized beds in recent publications [1, 2, 3, 4, 5]. It appears, however, that no serious a t t e m p t is made to relate the mixing coefficient to the local or microscopic characteristics of particle m o t i o n in any of these publications. In this work, a stochastic or statistical model has been proposed for the solids mixing in a liquid-solid fluidized bed, in which the particle velocity distribution and its variation are taken into account. Specifically, the particle diffusivity under the steady fluidized state has been derived by com-

bining the well-known Ornstein-Uhlenbeck equation of stochastic processes and the socalled partition function of particle velocities. The values of particle diffusivity calculated from the model are compared with the available experimental data.

CHARACTERISTICS OF PARTICLE MOVEMENT In this work, the movement of particles in a liquid-solid fiuidized bed is assumed to be diffusive in nature. Macroscopically, this movement in the system under consideration is steady, i.e. time-homogeneous; microscopically, however, the velocity of a single particle at any m o m e n t fluctuates in a random fashion. Suppose that the mixing process of particles in the bed consists of the superposition of a countless number of microscopic diffusive relaxation processes, each of which is represented by the 'Ornstein-Uhlenbeck' process as illustrated in Fig. 1. Furthermore, the initial condition of this transient process in terms of the instantaneous velocity of a particle is given by the so-called velocity partition function obtained by means of a statistical mechanical m e t h o d of the MaxwellBoltzmann type.

Partition function o f particle velocities Under a steady state condition, the average velocity of particles relative to a fixed coordinate or the wall of a batch fluidized bed is obviously zero. Suppose that the average particle velocity in the gravitational or z-direction relative to the upward superficial velocity of the fluidizing medium Ur0 is given by Uz. Then Uz=O--Uro

*On leave from Department of Chemical Engineering, Tokyo University of Agriculture and Technology, Koganei, Tokyo 184 (Japan). 0032-5910/85/$3.30

or

Uz=--Uf0

(1)

Furthermore, notice that the velocity of an individual particle in the fluidized bed fluc© Elsevier Sequoia/Printed in The Netherlands

146

of determining the partition function of particle velocities by resorting to the m e t h o d of statistical mechanics [6], we define

Qni

U~zi - Uzi -- Uzc

(4)

where Uzc is the minimum value of Uzi. For simplicity, Uzc is assumed to be fixed under a given fluidizing condition. Let

{A) NI

(5)

Nj = N 1=1

time (B) Fig. 1. Graphical interpretation of the superposition of microscopic diffusive relaxation (B) of movement of individual particles (A).

where N is the total number of particles, and Nj, j = 1, 2, ..., n, represents the number of particles at ~the jth state of particles, which has a velocity of U~i. Then, we can define

j=l

tuates constantly as stated previously, and thus it is usually different from Oz. Let Uzi be the instantaneous velocity of an individual particle in state j in the gravitational or z-direction relative to Uf0. Then, the instantaneous fluctuating velocity relative to Uz, denoted by U±si, can be defined as

U+~i = Uzi -- Uz

Uzi <~ Uz

N i. U~ = N. ~

Under the condition of eqns. (5) and (6), the partition function of particle velocities, P(U~ir), can be obtained by means of the Lagrange multiplier method, which yields [6]

Nj ~_ P(U~i) dU~i N

(2) -

or

U-~i = Uzi -- Uz

U~i > Uz

(3)

where U+~i is always a nonpositive (0 or negative) quantity, a n d U_si is always a positive quantity (see Fig. 2). The subscripts s is a new or transformed co-ordinate parallel to z; +s indicates the negative (upward) direction and --s indicates the positive (downward) direction on the z~o-ordinate. If the instantaneous velocity has no minimum limit, all the fluidized particles will be entrained from the bed. Thus, for the purpose N2

N3

1--?tUz,/9 9 " " O-z-U/s[-[~] ........

N

,,

Nn_ 1 N n

?' lUz°-i

,T/~Jz]--~~n~'~-~-Uzn

U-s'z2 Fig. 2. Relationship between the instantaneous fluctuating velocity relative to Uz, U+_si, and the instantaneous velocity of an individual particle relative to Uf0, UM.

exp --

~--~ dUz~i

(7)

Substitution of eqn. (4) into this expression gives

P(Uzi -- U~c) d(Uzi -- Uzc)

1

Uz -Uzc

exp(

~zz • ~czc] d ( U z i - Uzc) (8)

or

P(Uzj) dVzj _

N~

(6)

1 exp( Uz -- Uzc

Uzj--Uzct -~ - - ~z¢ / dUzi

(9)

Ornstein- Uhlen beck processes When the behavior of randomly moving particles in a liquid-solid fiuidized bed is characterized by the Langevin equation, the probability that a particle with the initial velocity, U±sjO, at time to will have the velocity U±sj at time t, f(U±sj, tlU±~j0, to) is repre-

147

sented by the Fokker-Planck equation, i.e. [7,81

Of(U+sj,tlU+~jo,to)

E[U±s2] = U+_sloexp(--aT) and Var[U±~j] = b[1 -- exp(--2aT)]

3t

(15) (16)

Let the particle diffusivity D~ be defined as

O2B(U±s~, t)f(U+_~jt]U+_~o, to) ~U±~j2

1 D , = - - Var[U±si] 2a

(17)

Because of eqn. (16), eqn. (17) leads to

~A(U+sj, t)f(U+sj,tlU+_sjo,to)

(lO)

-

0U+~y where U+_slis defined in eqns. (2) and (3). Suppose that the following conditions are satisfied at any instant in the system under consideration [9]

A(U+~j,t) =--aU+~

(11)

b [1 -- exp(--2aT)] Dp =

(18)

Substituting this expression, in turn, into eqn. (14) yields

f(U+~j, tlU±~jo, to) dU+~j 1

and

B(U+sj, t) = ab = constant

2a

•V/-•p a

(12)

X expl--

[ U+_si-- U±sio exp(--aT)] 2 4Dpa I dU±sj

where a is the inverse of relaxation time, and b is a parameter related to the variance of particle velocity; they are constant. Equations (11) and (12) together with eqn. (10) or the condition that both a and b are constant gives

Macroscopically, the system under consideration is of the steady state and thus E[ U+sj] in eqn. (15) is zero. Then, eqn. (19) reduces to

3f(U±sj,tlU_+~jo, to)

f(U±,s , oo) dU±~j

(19)

Ot

1

= ab O2f(U±~J' t]U+_~jo, to) 5U±~j2 +a

(13)

3U±~i

A stochastic process represented by this equation is the well-known Ornstein-Uhlenbeck process. Solution of eqn. (13) yields

[6,91

f(U+~j,tlU+_~jo,to) dU+~s 1

X exp --

2b[1 -- exp(--2av)]

(U+sj~ 1 4Dpa] dU+_si

(20)

f(U+~j, oo) dUz]

_

1

4~x/_~D~p a exp

{ (Uzj--Uz)21dUzj (21) 4Dpa

The particle velocity Uzj in eqn. (21) is defined in the range between 0 to ~. In a liquid-solid fluidized bed, however, the minimum of Uzj is Uzc, and thus eqn. (21) should be modified; the resultant expression is

V/2~b [1 -- exp(--2ar)]

[U+~j-- V+~jo exp(--ar)]2

exp

Combination of eqns. (2) and (20) yields

OU+_J(U+_~j,tlU+_sio,to)

l

~ a

~)(Vzj) dVzj

t

~ dU+-sY

- ~

(14) where

"c=t--to The mean E[ U+~i] and the variance Vat[ U+sj] of U+sy are given by, respectively,

exp

(Uzj Uz)2}} dUzj (22) 4Dpa )

where 1

k=

f ~(uzj) dU~j

Uze

(23)

148 DERIVATION OF PARTICLE DIFFUSIVITY

(A)

In a steady state system, the n u m b e r flow rates (cross-section area × flux) of particles in both +s- and --s-directions, F±s,are identical. The particle diffusivity can be estimated from this observation. From eqn. (22), the average particle velocity relative to the column wall in the descending direction, U_s, is given by

~

; Uzic~(Uzj)dUzj U_.~ = v~

(24)

; ¢(Uz~) dUzi v~

Uz

Uzj

.,,~_.(B)

~

Uzj

•

Substitution of eqn. (22) into this expression gives U-s = Uz +

--

(C)

(25)

Similarly, from eqn. (22), the average particle velocity relative to the column wall in the ascending direction, U++, is given by Uz

(26)

Vz

¢(Uzj) dUzj

f

Uzc Combining eqn. (22) with this expression leads to vZDpa[1 -2

--

exp(--y12)]

(27)

(D)

I

-U_$

U+ sj"

U-sj

)

Fig. 3. Relationship between the average particle velocity relative to the column wall in the ascending direction U+s, and that in the descending direction U-s- (A), Partition function of particle velocities, P(Uzi); (B), particle velocity distribution, (Uzi); (C), velocity distribution of U+sj; (D), distribution of

U-si.

direction, F+s, in such a bed. Since F_s and F+ s are given, respectively, by

erf(--Ym)

F-s = O_sS(AN) f P(Uzj) dUzj Vz

where

(29)

and

Uzj -

Y-

I

U+$

•

f Uzj¢(Uzi)dUzj U+s = Vzc

~1

2X/~Pa

t

(28)

-

F+s = U+sS(AN)

P(Uz~)dUz]

(30)

Vzc

Y1 -= 2 X,/~pa The relationship given in eqns. (25) and (27) is schematically shown in Fig. 3. The average velocity partitioned to individual particles in a steady fluidized state is identical to Uz. As stated earlier, the n u m b e r flow rate of particles relative to the wall column in the descending direction, F_+, will be equal to the n u m b e r flow rate of particles relative to the column wall in the ascending

i

z

where S is the cross-sectional area of the bed and A N is the n u m b e r density of the particles. Since F_+ = F+s

(31)

we have

v-, f P(Vzi)d zi = V++f°zP(Uz) dVz [Tz

~Tzc

(32)

149 Inserting eqns. (25), (27) and (9) into this expression yields - -2

~o GLassbe]ds ' j /J''' 5.0-Water System / / /

+ 1 -- exp(-- Yx2) ( e - - 1 )1 V/~pa 2

erf(--Y,)

= (e -- 2)Uz

(33)

Naturally the particle diffusivity Dp can be calculated from this expression, provided that the values of Uz and Uzc are known.

/

///

0.5 ~

/

///

Particle diffusivity based on the model The relationship between the superficial velocity Ufo and the minimum of Uzj, which is Uze, is required to estimate the particle diffusivity Dp from eqn. (33). This relationship, shown in Fig. 4, was determined empirically as a function of the superficial velocity Uro in previous work [10]. The particle diffusivity is defined as half of the square of particle displacement during a unit time interval [9, 11]. The particle diffusivity, calculated from eqn. (33), is a function of the inverse of relaxation time a and thus for convenience it has been assumed that ( l / a ) is the average time interval required for the velocity of each particle to reach from U±sio to the average diffusion velocity, i.e. the square-root of the variance of particle velocity, and that this time interval is considered to be approximately 1 s in the system 3.5

3.O

Glassbeads

1'02.0

2.5 '

Partlc[e Diameter,clp b 0.0543 c 0.0420 3.0 3i 5 40 ' i Uf0 Ccm/sec)

//

o b

o.ozz2

o0.0543 o,?

5.0 Ufo Ccrn/sec3

RESULTS AND DISCUSSION

/JJ

'c e Dameter Clp

4.5

Fig. 4. Relationship between the minimum of Uzi, Uzc, and the superficial velocity of the fluctuating medium, Ufo.

-

'

10

Fig. 5. Particle diffusivity calculated from eqn. (33) with the particle diameter as the parameter. under consideration (in contrast, the time constant for circulatory or bulk particle m o t i o n is substantially larger than 1 second). The resultant particle diffusivity is shown in Fig. 5 with the particle diameter as the parameter.

MODEL VALIDATION The values of particle diffusivity calculated from the present model under the condition of steady fluidization are compared with the available data in Fig. 6. The assumption is made that the effects of the distributor and outer walls on the velocity distribution of particles are negligibly small. When the velocity Uz or --Ufo is relatively low, the particle diffusivities based on the model agree well with the available experimental data [1, 2, 3, 4, 12]. However, some discrepancy has probably arisen because when the superficial velocity of the fluidizing medium Ufo is high, the experimental data would deviate appreciably from the Fickian diffusion equation. This is based on the existence of circulatory particle flow in a liquid-solid fluidized bed. Almost all the particle diffusivities obtained by other investigators have been estimated by fitting the experimentally determined profiles of particle concentration to the solutions of the Fickian equation. Typical data and results discussed in this section are summarized in the Table. Sample calculations are also quoted in the table.

150

0 X X ~ oo 4.~

T0 X ~

O0 ~ 0

~

0

~

~

~

0

0

~ E'-.-

~

70 X o 0

•

~o

0

0

0

0

0

0

0

0

0

0

0

0

0

¢0 cO

~

0 0 0

0 0,1

c~ II

? 0

M d ~ 4 ~ 4 ~ M N ~ d ~

co

'-'~ 0

o

~

.o X

o" 0 ,-~

~D

CO

0

0

t~ ~

0

0 0

2 ~'~

•~

~

-~

0

,.~

,.~ ~

0

,.~

- -

0

0~

0 ,.4

~

0

•

X

~

,..0

o~

151 Investigator (yeor),Moteriol,Dp{cm), Liquid, Key i

Carlos,

Glass beods,O.9

i

v

i

I

Di m e t h y l - ~ .

et al. 1968) phlhalote. l..k:mdle¥ , Soda Glass beod.%Q09 Methyl etal. 1966) benzen 0.1539 beads, Q0952

Keflned~/, Glass el al. 1 9 6 6 ) M~hi

et?~l.(t961) M a r b l e

Lime

-O-

0,0841 Water

•

0.0681

Ill

0.0555

n

stone 0.0841

A • v

0.06 81 Fire brick 0.0841 Yutani

0.0681 • .=t?l.(1982)Glass b e o d s O . O 7 7 2 W a t e r O

o.o54s

0.042o

. . . . . .

"20

-I0

5

/ /

•

e .a"^_~ ~ ~,-~uqb o ' / ~ i ~ ,

L)

ou3 2

/~o~

~

'~o

U

x

Cb v

Q3

o.~

i

I

i i

i

i

i

i

= L | i

Lo ~.o s.o ( Dp)cal. Ccrn2/sec)

~o

30

Fig. 6. C o m p a r i s o n of t h e values o f particle diffusivity based o n t h e p r e s e n t m o d e l w i t h available data given b y o t h e r investigators.

B(V±sj, t)

CONCLUSION

The following significant results have been obtained in this work. First, the particle diffusivity can be estimated based on the proposed model. The values of particle diffusivity obtained from the model agree well with those experimentally obtained by other investigators if the superficial velocity of the fluidizing medium is relatively low. Second, the solids-mixing process in the bed can be modeled by superimposing on each other a countless number of microscopic, diffusional relaxation processes of particles, which are time-dependent. This provides a logical interpretation for the discrepancy between the values based on the model and those obtained by fitting the Fickian diffusion equation to the data.

LIST O F S Y M B O L S

A(U±sj, t) a

parameter defined by eqn. (11) inverse of relaxation time

parameter defined by eqn. (12) variance of the particle velocity particle diffusivity particle diffusivity experimentally obtained (Dp)ca 1 particle diffusivity calculated from eqn. (33) E[U±sj] mean of U±sj F+~ number flow rate of particles in the +s-direction F~ s number flow rate of particles in the --s-direction F±s number flow rate of particles in the +s-direction f(U±~j, t[U+~jo, to) or f(U±~i, t) probability that a particle with the initial velocity U±~io at time t o will have the velocity U±~i at time t f(U+si, oo) probability that a particle will have the velocity, U±~i at time k constant given by eqn. (23) N total number of particles Nj number of particles at the ]th state of particles b Dp (D,)ex p

152

P(U~j) t to Ur0 U-bS

~Y-s

U_~

ZC

u~j

~j Var[U±~j] W Y Y1

partition function of particle velocities arbitrary time initial time superficial velocity of the fluidizing medium average particle velocity relative to the column wall in the ascending direction average particle velocity relative to the column wall in the descending direction instantaneous fluctuating velocity relative to Uz, defined by eqn. (2) instantaneous fluctuating velocity relative to Uz, defined by eqn. (3) average velocity of particle relative to the upward superficial velocity of the fluidizing medium minimum value of Uzi instantaneous velocity of an individual particle relative to the upward superficial velocity of the fluidizing medium velocity defined by eqn. (4) average value of U r] variance of U±~ number of w a y s arbitrary variable defined by eqn. (28) dimensionless number given by eqn. (28)

Greek symbols ~(Uzi)

constant constant particle velocity distribution function of Uz] given by eqn. (22)

t -- t 0, time interval

Subscripts c

J

m

s

critical or minimum value jth state upward direction, i.e. the opposite to the gravitational direction downward direction, i.e. the gravitational direction z-co-ordinate; positive in the gravitational direction

REFERENCES 1 I. Muchi, S. Mukaie, S. Kamo and M. Okamoto, Kagaku Kogaku, 25 (1961) 757. 2 D. Handly, A. Doraisamy, K. L. Butchier and N. L. Franklin, Trans. Instn. Chem. Engrs., 44 (1966) T260. 3 S. C. Kennedy and R. H. Bretton, AIChE J., 12 (1966) 24. 4 C. R. Carlos and J. F. Richardson, Chem. Eng. Sci., 23 (1968) 825. 5 E. Ruckenstein, Ind. End. Eng. Fundamentals, 3 (1964) 260. 6 F. Reif, Fundamentals o f Statistical and Thermal Physics, McGraw-Hill, New York, 1965. 7 N. T. J. Bailie, The Elements o f Stochastic Processes with Applications to the Natural Sciences, Wiley, New York, 1964. 8 D. R. Cox and H. D. Miller, The Theory o f Stochastic Processes, Chapman and Hall, London and New York, 1980. 9 E. Nelson, Dynamical Theories o f Brownian Motion, Princeton Univ. Press, Princeton, NJ, 1967. 10 N. Yutani and N. Ototake, Kagaku Kogaku Ronbunshu, 6 (1980) 570. 11 C. Houghton, I. & E.C. Fundamentals, 5 (1966) 153. 12 N. Yutani, N. Ototake, J. R. Too and L. T. Fan, Chem. Eng. Sci., 37 (1982) 1079.