A modified coalescence—dispersion model for the axial mixing of segregating particles in a motionless mixer

Powder Technology. 16 (1977) 217 - 231 0 Elsevier Sequoia S-A., Lausanne - Printed in the Netherlands

A Modified CoalescenceDispersion Motionless Mixer

Model

217

for the Axial Mixing

of Segregating

Particles in a

L. T. FAN and Y. CHANG Department (Received

of Chemical February

Engineering,

19, 1976;

Kansas State University.

in revised form August

SUMMARY

A modified coalescence-dispersion model has been developed for the axial mixing of segregating particle systems in a motionless mixer (Kenics mixer). This model is capable of generating the concentration distribution as a function of time and its applicability is not constrained by the initial concentration distribution in the mixture. Two important parameters in this model are the coalescence rate and the distribution ratio. It has been found that the former can be correlated as a linear function of the number of helices in the mixer, and the latter depends heavily upon the physical properties of the individual particles. The validity of the present model has been tested against the available experimental data [lS] _

INTRODUCTION

Solids mixing is a common processing operation widely used in a variety of industries. It is extensively employed in the manufacture of ceramics, fertilizers, detergents, glass, pharmaceuticals and animal feeds, and in the powder metahurgy industry. Three principal mechanisms are known for the process of solids mixing: convective, diffusive and shear ]I] - However, the extent of mixedness as a’function of time is generally difficult to predict because satisfactory theories capable of describing the mechanisms have not been fully developed. The phenomenon of solids mixing has been analyzed based on essentialIy linear diffusion models [l - 4]_ Such a simple approach is applicable only to nonsegregating particle systems, i.e. highly idealized systems.

Manhattan.

KS 66506

(U.S.A.)

19, 1976)

The behavior of systems where the particles differ in properties and characteristics (segregating particle systems) is a matter of practical concern in industrial operations. The occurrence of demixing (segregation) is extremely important, since it can markedly alter product quality. Despite this importance, the simultaneous occurrence of mixing and demixing in segregating particle systems has received relatively little attention. Some attempts have been made to formulate mixing rate expressions for segregating particle systems. Rose 153 derived his expression by incorporating the so-called mixing and demixing potentials into a regular diffusion expression_ While his model has been successfully employed in representing some experimental data, definition of the demixing potential is conceptually difficult to visualize. It was pointed out [S] that the estimated parameters in the model are inconsistent with the theory behind the model. Weydanz 173 proposed a model for a two-dimensional process in which the segregation effect is present in only one dimension. The main assumptions were that the rate of segregation is constant and that the volume interchanged in either direction is constant_ Neither of these assumptions can be espected to be valid in anything other than highly limiting cases. Faiman and Rippie [S] suggested that both mixing and demixing are first-order processes and that the process may be considered analogous to a reversible chemical reaction in which both forward and reversible steps occur simultaneously. The rate expression thus derived can only describe the later stage of a mixing and demixing process. Fan et al. [9], assuming that mixing and demixing can be described independently, presented a mechanistic model which is analogous to

21s

chemical kinetics of two complex reactions in series. The model can describe mixing curves (degree of mixedness versus time plots) to some extent; however, it gives no concentration distributions which are estremety important in the design of multiple misers and continuous mixing systems_ A modified coalescence-dispersion model has been developed in the present study. The

DESCRIPTION

OF THE

concept of coalescence-dispersion was introduced by Curl [lo] and has been employed by many investigators to study the effect of dispersed phase mixing on various reactions [ 11 - X5]_ The validity of the present model has been tested against the available data 1161. ReIationships between parameters of the proposed model and physical properties of the system have been analyzed.

MODEL

In the present model, the entire packed mass of a misture is equally divided into a finite number of sections in series in the vertical direction. Each section in turn IS equally divided into a finite number of cells in the horizontal direction_ Each cell contains a specified number of particles- Schematic representation of the ceIIs and sections is given in Fig. 1. Mixing particles of the same size which may or may not have other similar characteristics is considered fiit_ The following mixing mechanisms are assumed (see Fig_ 2 j: (1) The convective mising in the asial direction is induced by allowing random coalescence between cells in the adjacent sections, namely the ith and the (i + 1)th sections, two at a time. Ko coalescence is possible between cells in the same section. The number of particles in each cell and the number of coalescences per unit time (coalescence rate) are invariant with respect to time and position_ (2) The diffusive mixing mechanism is induced by collision and redistribution of individual particies between coaIescing ce1I.s. Upon each coalescence, particIes in one coaIescing ceI1 experience one-to-one random collisions with particles in the other coalescing celi and redistribute

I

2

1

Fig. 1. Schematic representation particles In the mixture mass.

l-n-l

ofsections,

m

cells and

Fig_ 2. Representation of the visualized of miring upon a coalescence_

mechanism

219 themselves instantaneously between the two coalescing cells. In the tease of a key-non-key particle collision, the key particle may find its new position in the coalescing cell either in the ith section or in the (i + 1)th section_ The probability of the key particle to fiid its new position in the ith section is denoted by LY,and consequently that in the (i + l)th section is denoted by (1 - a)- The magnitude of the distribution ratio Q is mainly dependent on physical characteristics of the particles. When the key and non-key particles are identical in every aspect except color, Q is equal to 0.5. (3) The two coalescing cells are redispersed into their original positions in the respective sections. Based on these assumptions, for an arbitrarily small interval At, we have (see Appendix 1): The number concentration of key particles in cell j in the ith section at t + At provided that the coalescence is with cell k in the (i + 1)th section, Ci.;(t + At) is Cci.i(t)Ci+I_k(t)

+ CC1 -ci.j(t)l

Ci+l.k(t)

+ Ci.j(t)Cl

-cc-

z+~.k(t)IIaI

X CJAt

+ C,.j(f)(1

--At) (1)

The number concentration of key particles in cell k in the (i + 1)th section that the coalescence is with cell j in the ith section, Cj+ l&(t f At) is [Ci.j(t)Ci+l.k

(t)

Citl,k(t)(l

+{I1

-

-ci.j(t)l

Ci+l,k(

i? ) + Ci.j(t)[l

i=l,2,...,n-1

UAt),

-Ci+l.k(t)])

at t + At provided

X (1 --LY)]

UAt

+

j, k = 1, 2, .__, m

(2)

where Ci.j(t) = number concentration of key particles in cell j in the ith section at t, Ci+ I.c(t) = number concentration of key particles in cell k in the (i + 1)th section at t, and U = number of coalescences per unit time_ Mixing narticles of different sizes is complicated by the fact that the total number of particles in a cell is :! function of the composition and the structure (packing) of the mixture in that cell. The great difficulty involved in such systems can be overcome by assuming that replacement of a key particle lost from the cell can be accomplished either by another key particle or a number of non-key particles provided that the key particles are larger than the non-key particles_ Under this circumstance, the assumption of one-to-one particle collision is no longer valid unless a certain number of the non-key particles is treated as an aggregate, which behaves as if it is a single particle. For this purpose, the apparent pumber concentration of key particles in cell j in the ith section is defined as [Ci.i(t)l,

=

Xi.j(r) xi*i(t)

yi i(t) + L r

’

i = 1, 2, ___, n

j = 1, 2, ___, m

(3)

in which Xi-i(t) = number of key particles in cell j in the ith section, Yi.j(t) = number of non-key particles in celli in the ith section, and r = bulk volume ratio of a key particle to a non-key particle. With this definition, eqns. (1) and (2) are applicable to the mixing of different size particles.

SYSTEM

EQUATIONS

The probability that cell j in the first (top) section and cell k in the second as the two cells of a coalescing pair (see Appendix 2) is 1 nm2 ’

section

are selected

j,k = 1, 2, ___, m

The probability that cellj in the ith section and cell k in the (i + 1)th section two cells of the coalescing pair (see Appendix 2) is

(4) are selected

as the

220

1

i = 2,3,

2nm*’

___, n -

1

j.k = 1, 2, ___, m

(5)

The probability that cell j in the nth (bottom) section and cell k in the (n - 1)th section are selected as the two cells of the coalescing pair = the probability that cell j in the first section and cell k in the second section are selected as the two cells of the coalescing pair, Le. 1

j-k = 1, 2, _._, m

nm’)

(6)

The change in the number concentration of key particles in cell j in the itb section during At due to the coalescence, provided that the coalescence is with cell k in the (i + l)th section, is ICi.i(9G-,_k ci.j(r)

(t) +I[1

(1

-

u

ac)

-cci.i(f)l

-

Cicl_k(t)

f

i =

ci.j(t)*

ci.j(t)[l

-ci+l.k(t)lbl

X

j-k

1, 2, ___) .x- 1

uat

f

= 1, 2, ___, m

(7)

The change in the number concentration of key particles in cellj in the ith section during At due to the coalescence, provided that the coalescence is with cell k in the (i - 1)th section, is ICi.;(t)Ci-r_,(t) ci.,(L)(l

-E-I[1 -Cr_i(t)l

-

uAt)

-

Ci-r_*(t)

+ C,.i(t)[l

The change in the number

concentration

= 1, 2, ___, m

j,k

i = 2, 3, ___, n

ci.j(t),

--(r)l X UAt +

-ci-l.k(t)]](l

(8)

of key particles in cell j in the first section

during

_u is

c,,tt f

At) - cl.,(t)

1

=-

2

nm2

UAt = “_ rrm

ECCl_i(~)C~_tz(t) + CCl - Cl_j(r)I

k=I

m c k=l

Ec,.J(t)c,.k(t)

+

f[l

-

cl.j(t)]

C2.k(t)

c2.k

tt)

+

+

- - c,,(t)]}x

Cl.i(t)Cl

cl.i(t)E1

-

G.k(~)llX

a]

Q -

U4t

+

C,.j!Ol

D

j = 1, 2, ___)m

(9)

The change in the number

UAt = 2nm’

_g CCdaCr-1 R I

ci.j(t)

f

i=2,3

Ci.j(t)Ci+l.k

, .. .. n -

.k (f) +ICl tt)

=z

f at)

of key particles in cell j in the ith section

-Ci.j(t)]

fCC’--ci.i(t)l

Ci-l-k(t)

+

Ci_i(t)

[I

-CC-l-k(t)]}

concentration

during At is

(1 --a)

-

ci+l&(t) +ci.i(t)C1-Citl.k(t)l}Q-Ci.i(t)]r

j = 1, 2, ___, m

1

The change in the number C&t

concentration

(10) of key particles in cell j in the nth section

during At is

-C&t)

Z

ICdtiG-,k

.

(f)

iECl

k=l

G&j(t)1>

j = 1, 2, .__, m

-Cn.j(t)I

Cn-l.,(t)

tCn.j(t)

Cl -cn-l.k(~)II(l-~)-

(11)

221

Dividing

eqn. (9) by At, and setting At *

0, we obtain

d&(t) dr

z s

=

rm2

Ccl.j(w2.k(t)

-I- {Cl

--l.j(t)l

c,,(t)

+ cl.;(t)

: a - cl.i(t)l,

Cl - c*.,(t)1

k=l

, ___, m

j=l,2

(12)

Similarly, dG.i(C dt

u

=-

jE Cci.i(t)c-

2nm’

c=l

L-l.&(t)

ci.j(t) + Ci_j(t)Ci+l_k i = 2, 3, ___, n -

+{I1 -ci.j(t)l

(t) + II1 -ci.i(t)l 1

Ci-l_k(C) Ci+l.k(t)

+

-I- Ci.,(L) ci,j(t)[l

[l -cci-l.i;(t)](l -C~+*,k(t)]}(Y

--(Y)

(13)

-Cci,j(t)],

j = 1, 2, ___)m

and d%(t) dt

=

A!s nm2

cc,,(t)c-I&)

-I- EC1- cn.i(t)l

c,-,_,(t)

+ &j(t)

cl - c,z-l.k(t)l

I(1 -a)

-

kzl

ctkj(t)l

j=l,2

Y

, ._., m

(14)

Equations (12) - (14) can be solved numerically to give the number concentration of key particles in each cell. Note that, in reality, we only need to solve simultaneously n equations instead of nm equations because the governing equations for the cells in the same section are identical. When the model [eqns. (12) - (ld)] is applied to a mixer operated in a semi-batch mode, e.g. repeatedly passing a mixture through a motionless mixer, the time of mixing can be considered to be proportional to an integer number of passes when end effects are negligibly small. The proportional constant has been found to be 1.5 by comparison with the esperimental data. A more convenient discrete time approach is described in the next section.

MONTE

CARLO

SIMULATION

When At equals to l/U Ci_j

(i.e. period of a coalescence),

eqns. (1) and (2) reduce to, respectively,

( ’u> t +

=ci.j(t)ci+l.k

(15)

(t)+CC1-ci.j(t)]Ci+l.k(t)+Ci,j(t)[1-Ci+l.k(~)]}Q

and G*l.A

1

( ) t-f--

u

=ci.j(t)ci+l.k(t)

i = 1,2,

+CCl

__., n -

1

-Cci.j(t)I

CicI.k(t)

j, k = 1, 2, ___, m

+ci.j(t)

Cl

-Ci+l.k(t)II(l

--a),

(16)

Fig. 3. Flow

chart

for the Monte

Carlo

simulation.

where Ciei(t + l/U) and C,, I.* (t + l/U). by their definitions, are respectively the number concentration in celli of the ith section and that in cell k of the (i + l)th section after the coalescence. According to the Monte Carlo simulation technique [l’i, 181, the mixing process described by eqns. (12) - (14) can be simulated by using only eqns. (15) and (16) in the following manner (see Fig- 3): (1) -4 section is selected at random by generating a random number i1 where 1 =S i, =Gn. (2) X cell is selected at random from this section by generating another random number jr where 1 < il < m. (3) If the section selected in (1) is neither the top section nor the bottom section, one of the two adjacent sections is selected at random by generating a third random number i2 where 1 d i3_d 2- If the section selected in (1) is the top section, the second section from the top is always seiected, and if the section selected in (1) is the bottom section, the second section from the bottom is always selected. (4) A cell is selected at random from the section selected in (3) by generating a random number & where 1
EXPERIMENTAL

DATA

The procedures employed 1161 to obtain the data in this work are outlined below. The mixing device employed to obtain the data (Kenics mixer) consisted of a l-5 in. i-d. Pyrex tube with a number of helical elements. Experiments were carried out by three basic operations: loading, passing, and sampling_

In the loading process, key particles (see Table 1) were poured into the feeder first; particles of the second component (non-key particles) were carefully layered on top. After loading, the feeder was placed above the mixer and the gate valve was withdrawn rapidly, permitting the slug of particles to fall freely through the mixer. The particles were caught in the collector; then the feeder

2 6 12 2 5 12 12 G 8 12 6 7 9 12 $2 6 8 12 12 12 G 8 12

No, of hcllccs in the mixer

Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Glass Lucite Lucite Lucite l/8 l/S l/6 G/32 G/32 6/32 3116 l/6 l/8 l/8 l/S l/8 118 l/S 3/1G 3/1G 3/16 3/16 3/16 3/16 5132 6132 5132

1.18 1.18 1.18 1.18 1018 1,18 l-18 1,lS 1,1&i 1,18 1,18 1.18 1918 l-18 1.18 1,18 1,lA 1018 1-18 2.42 1,18 1,16 1818 hcitc Lucilc Lucite Lucite Lucite Lucilc Lucite Lucite Lucite Lucite Lucite Lucite Lucite Lucite Glnss Glnss Glass Glnss Slccl Steel Glnss Glnss Glnss l/8 l/8 l/8 6132 G/32 5132 3/1G 3/16 3/16 3116 5/32 6132 5132 5132 3/16 3/16 3/16 3/1G 3/1G 3/1G 3/lti 3/1G 3/16

Pnrticlc size (in,)

Mntcrinl

Density Wma)

Mntcrinl

Pnrticlc size (in,)

Sccontl component

Key com~oacnt (charged in the bottom)

1.18 1.18 1018 1,18 1.1t-I 1,lS 1.18 1,18 1,18 1,lA 1018 1.18 1,18 1,lS 2.4 2 2.4 2 2,42 2,42 7,GO 7.60 2,112 2.4 2 2,42

Dcusity (dd) 0,500 0,500 0,500 0,500 O.GOO 0,500 0,515 0,680 0,680 0.680 O.GOO 0.690 O.GOO 0,600 0.515 0.516 0.51G 0.51G 0.52 0.50 0.68 0.58 0.58

Fraction of key component

Estimated values of conlesccncc nntl dibtribution parnmclcrb bnsctl on cxpcrimuntd tlnln by Gclvcs~Arocl~~[ lti]

TABLE 1

48 113 253 35 64 121 18 56 17 116 69 85 104 128 22 63 66 100 184 105 51 64 115

Conlcsccncc rntc, II’

II - 9, 111= 10

0,GOO 0.600 0,600 0,600 0.500 0.600 0,600 0,726 0,773 0,797 0,715 0,726 0,747 0.762 0,173 0,777 0,875 0,927 0,969 0,958 0.886 0,891 0.951

Distribution mlio, (Y

and the collector were interchanged to carry out the second pass. This process was repeated to give the desired number of passes. To determine the fluctuation of the concentration of the key particIes in the mixture mass, the variance was caIcuIated by using the equation (17) where c denotes the mean concentration of the key particles in the mixture mass and n represents the number of samples taken from the mixture. For the present case the entire misture was divided into samples. The degree of mtsedness was calculated for each experiment according to the definition

08.

-

ElpenmaIO

----

Epuoclmru2l-u4~

dolo

-----

Nnt.z

cab

---_-

me

s

z

OS-

pz-

E

z =

04.

z

1

1

----I

A

I

----

I i

I

I

IL3SCF7S~ Cell

numt!cr

Fig. 4(n)_ Concentmtion profile for mixing ‘/IS in_ glvs particles with 3/~6 in. steel particles after 2 passes in the 12-helices mixer.

where a$ is the variance of the completely segregated state of the mixture, which can be cakulated by o;=E(l-Ec,

RESULTS

(19)

AND

DISCUSSION

Computations for solving eqns_ (12) - (14) for 9 sections by means of -4dam’s method have been performed on an IBM 370 computer. Simulations by means of the Monte Carlo method have also been carried out by dividing the mixture mass into 9 fictitious sections in the vertical direction with 10 cells arranged horizontally within each section. TypicaI results are shown i=i Figs. 4(a) - (e), wNch give concentration profiles for the axial mixing of %a in. glass beads with the same size steel baIIs_ Good agreement between the esperimentaI data and the two simulations can be observed in general_ The over& progress of the mixing and demixing process can also be illustrated by a plot of the degree of mixedness M uersus the number of passes as shown in Fig_ 5. As eupetted, the simulated result based on eqns_ (12) - (14) is deterministic in nature while that based on the Monte Carlo simulation is stochastic in nature_ In this aspect, the Monte Carlo simulation appears to be a better representation of the process- The difference in the two simulated results is not significant at the

Fig_ -l(b)_ Concentration profile for miring 3/16 in. glass particles with 3/16 in. steel particles after 4 passes in the l%helices mixer.

‘OC

Fig. 4(c)_ Concentration profile for mixing ‘/~a in. glass particles with 3/16 in. steel particles after 6 passes in the l2-helices mixer.

1.0

08

f

08

2 5

i;

04

= f %

02

00

Fig. 4(d). Concentration glass partiCkS With ‘/16 in the IZ-helices mixer.

b----. _____ i------.-

1

profile for mixing 3/~6 in. particles after 8 passes

iII_ Steel

-

-msmt~a

---

~um.¶luu2l-osl

-----

uaue

culomellmd

2

a dynamic equilibrium state was reached and no further improvement in the homogeneity may be obtained. The rate of mixing was strongly dependent on the particle size. Mixing progressed most rapidly when the particle size was small. The progress of mixing for this case can be described by the proposed model with (Y = 0.5 as shown in Fig. 6. The values of the coalescence rate (number of coalescences per pass), U*, estimated from Fig. 6 are presented in Table 1 and plotted against (l/particle diameter)3 as shown in Fig. 7. A near straight Iine can be obtained from the plot, indicating that an important factor for mixing is the volume of particles. The line passes through the origin when extrapolated. This implies that as particles become excessively large and therefore (l/particle diameter)3 excessively small, no mixing can be achieved in the _n-Gxer. The result is intuitively expected, as mixing is limited by the dimension of a mixer when the particles mixed are estremely large_

Fig. 4(e)_ Concentration profile for miring ‘/M in. glass particles with 3/x6 in. steel particles after 10 passes in tbe 12-helices miser.

5% confidence level. However, the computing time needed for the simulation based on eqns. (12) - (14) is about twice as much as that needed for the Monte Carlo simulation. Simulation for all other particle systems has been performed by using the Monte Carlo simulation. The degree of fluctuation in the Monte Carlo simulations is dependent upon the number of cells identified in the mixture mass. It can be reduced by increasing the number of cells at the expense of more ccmputing time. Nevertheless, 90 cells in the mixer have been found to be sufficient in this study. -Nonsegregating particle systems Experimental data [lS] in Fig. 6 show that the rate of mixing was relatively fast initially and decreased as the number of passes increased- After a sufficient number of passes,

Fig. 5. Degree of mixedness OS. number of passes for mixing %6 in. glass particles with %a steel particles in the 12-helices mixer.

Fig. 6. Effect of particIe size on the rate of miring for ntinsegregating particle system in the 12-helices mixer_

226

a terminaI value corresponding to a dynamic equilibrium state (see Fig. 9). Estimated coalescence rates presented in Table 1 can be correlated by U’ = 22 f 9 X (number for

l/8

-

731

in. Lucite particIes,

U’ = 10 + 9 X (number .’

for

:

Fig. ‘7. Coalescence rate as a function of reciprocal of particle diameter to the third power for mixing aonsegregating particle systems in the 12-helices mixer.

I

I

or 0

z

‘

I 3

4

5&

JM_t

9

LO

II

Fig_ 8. Estimated coakscence rates of functions the number of helices in the mixer.

1.3

I2

of

The coalescence rate is plotted against the number of helices in the mixer as shown in Fig. 8, which indicates the following correlations: U’

= II+

20 number

of helices

for r/s in. Iucite particies, U‘ = 14 + 9 number

(20)

‘/8

of hehces)

- 3/16 in.

hcite

(22)

and

of heiices)

(23)

pZid&s.

The coalescence rate for the ‘18 - 3/16 in. lucite particle system was smaller than that for the */s - s/3= in. Iucite particle system, while the size difference for the former was greater than that for the latter. It is possibie that, in addition to size difference, the coalescence rate is also a function of the individual particle size. The size and the size difference may couple together in a comples fashion that cannot be easily described. The distribution ratio, CY,which appeared in the model, indicates the tendency of a particle system to segregate. While it is impossible to determine esplicitly the dependence of the distribution ratio on the size difference from the Iimited number of the available data, it appears that an increase in the size difference results in an increase in (Y_ Figure 8 shows that the linear relationship between the coaIescence rate and the number of helices is less obvious for the system with density difference than that for the system with size difference_ It may be that the end effects are more pronounced for the former than for the latter. The rate of mixing for various combinations of particles differing primarily in density is shown in Fig. 10, which reveals the fact that the greater the density difference the faster the mixing and

and

of helices

(21)

I,

for

5/32 in. lucite particles_ Strictly speaking, these equations are not valid for empty tubes and should be used for mixers with more than one he&_

Segi-ega

ting particle

systems

For the system with size difference, the smaII particles were initially pIaced at the bottom and the large particIes were layered on the top [lS] _ A typical degree of mixedness uerstcs time plot shows a rapid initial rise in M to a maximum, followed by a decline to

Fig. 9_ Effect of particle size difference on the rate of mixing and segregation in the 12-helices mixer.

227

mixes within a single pass. a/(1 - a) is plotted against Ap fp, in the same figure. which indicates the correlation a (1 -a)

Fig. 10. Effect of particle density difference on the rate of mixing and segregation in the 12-helices mixrr.

RO-

-S!J

no. l50-

= 1.1

+ 20.6

x Apip,

(24)

For the system with both density and size differences, the small particles were initially placed at the bottom and the coarse were Iayered on the top. The degree of misedness for this case is plotted as a function of the number of passes in Fig. 12. As can be seen from the figure, the rates of mising and demising are much greater than the two previous systems. The combined effect of size and density differences is greater than the single effect of either the size or density difference.

-25

pB¶_

Fig. 11. Estimated parameters of functiors of ApJp, for mixing particles of different densities in the 12helices mixer. .?

Fig. 12. Effect of particle size and/or density differences on the rate of mixing for various particle systems in the 12-helices mixer_

demixing. The estimated coalescence rate is plotted against Q/p, in Fig. Il. It appears that the rate of increase in the coalescence rate with Ap/p, is reduced as the density difference becomes smaller. After the density difference reaches a certain value, the increase in the coalescence rate becomes so drastic that eventually the particle system mixes and de-

CONCLUSIONS

AND

SIGNIFICANCE

A modified coalescence-dispersion model has been developed and successfully applied to the correlation of the available experimental data [IS] _ For the nonsegregating systems and the segregating systems with size difference, the coalescence rate is a linear function of the number of helices in the mixer. The linear relationship between the coalescence rate and the number of helices in the miser is Iess obvious for the system with density difference than for the system with size difference. The dependence of the coalescence rate on the size of particles indicates that an important factor for mhing is the volume of individual particIes for the nonsegregating system. The distribution ratio appeared to be dependent upon the size, the size differeixe, the density, and the density difference. While a previously proposed deterministic model [9] for mixing of segregating particle systems in motionless mixers contains four parameters, the present stochastic model contains only two parameters. Unlike the deterministic model, the present model is capable of generating concentration distribution as a function of time, and its applicability is not constrained by the initial concentration distribution in the mixer. Therefore the present model may be valuable in understanding the mechanisms of mixing in the motionless mixers as well as for the improvement of mixer design.

To derive a model which can take into account additional details of mixing mechanisms, such as the cell size distribution and the coalescence rate distribution in the mixer, further than can the present model, additional analysis and experimentation are required.

5!u

_4CKXOWLEDGEbIENTS

The authors wish to thank Dr_ M_ Siotani, 13r_ F_ S_ Lai, Mr_ R. H_ Wang and _Mr_ S. H. Shin for participating in many helpful discLlssions_ This work was supported by the National Science Foundation (grant ENG 73-04008A02).

LIST

OF SYMEOLS

c

ci G.i (ci. j), n-2

n

w r

u U’ xi.j

yi.j Y

P

u* 65 P Pm

I_ DERIVATION

(Al-l)

where U denotes the number of coalescences per unit time_ The number of collisions during an arbitrarily small interval At is (Al.2)

eu4t

The probability of key-key particle collisions, provided that celli in the ith section coalesces with cell k in the (i + 1)th section, is c,.i(t)Ci+l.k(f),

average number concentration of key particles in the mixture ave_rage number concentration of key particles in section i number concentration of key particles in cell j of the ith section apparent number concentration of key particIes in cellj of the ith section number of divided cells in each section number of divided sections in the mixture mass number of passes through the mixer bulk volume ratio of a key particle to a non-key particle number of coalescences per unit time number of coalescences per pass through the mixer number of key particles in cell i in the ith section number of non-key particles in cell i in the itb section distribution ratio total number of particles in each cell variance variance of the completely segregated state density average density

APPENDIX

be denoted by !2. Based on assumption (2), upon coalescence, each particle in one of the two coalescing cehs experiences a one-to-one collision with another particle in the remaining cell in a completely random manner. Thus, Q collisions occur for each coalescenceThe number of collisions per unit tune is

OF EQNS.

(1)

AND

(2)

According to assumption (l), the number of particles in each cell is a constant which is to

= 1. 2. ___~n -

i

j, k = 1, 2, . . . . m

1 (A1.3)

The expected number of key-key particle collisions during at, provided that cell i in the ith section coalesces with cell k in the (i + 1)th section, is Ccf.j(t)Ci i

-c-I.k(t)l

(Al_4)

(Quat),

= 1, 2, _._, n -

I

j, k = I, 2, ___, m Based on ;1s: umption (2), after each keykey particle collision, the two particles in the colliding pair split instantaneously and distribute randomly between the coalescing cells. Thus, the expected number of key particles in celli in the ith section resulting from keykey particle collisions during 4.t, prcvided that the co A escence is with cell k in the (i + 1)th section, is Cci.i(t)Ci+l_&(t)l i

Ceunt)

= 1, 2, __., n -

x 1,

(A1.5)

1

j, k = 1, 2, .__, m The expected number of key particles in cell k in the (i + 1)th section resulting from key-key particle collisions during At, provided that the coalescence is with cell k by the (i f l)th section, is CciJci i

* 1.k (t)l

wu40

= 1, 2, -__, n -

j-k = 1, 2, ___, m

x 1,

1

(A1.6)

229 The probability of key-non-key particle collisions, provided that cell j in the ith section c o a l e s c e s w i t h c e l l k i n t h e ( i + 1 ) t h s e c t i o n , is [1

--

i

= 1 , 2 , ..., n - -

1

{[1 - - C i . i ( t ) ] C i , l , k ( t

(AI.8)

2 ..... m

According to assumption (2), upon collision, a key particle will have a probability of a to find its new position in cell i in the ith section and a probability of (1 -- ~) to find its new position in cell k in the (i + l)th section. Thus, the expected number of key particles in cell ] in the ith section resulting from keynon-key particle collisions during At, provided t h a t t h e c o a l e s c e n c e is w i t h c e l l te i n t h e ( i + 1 } t h s e c t i o n , is

The e x p e c t e d n u m b e r o f key particles in cell j in the ith section which d o n o t collide during At, p r o v i d e d t h a t t h e coalescence is with cell k in t h e (i + l ) t h section, is Ci.i(t ) £(1 - - U At),

j=l,

=1,2

],k=l,

(A1.9)

..... n--1

The expected number of key particles in cell/z in the (i + 1)th section which do not c o l l i d e d u r i n g ,xt, p r o v i d e d t h a t t h e c o a l e s c e n c e is w i t h c e l l j i n t h e i t h s e c t i o n , is C~+ i . ~ ( t ) ~(1 - - U A t } , i=l,

2,...,n--i

] = 1, 2 , ..., rn The total expected number of key particles in cell ] in the ith section at t + At, provided t h a t t h e c o a l e s c e n c e is w i t h c e l l k i n t h e ( i + 1 } t h s e c t i o n , is

c i + 1..(t) + c~.j(t)

Ci + , ~ ( t ) ] } (QU A t ) (1 - - r.), 1,

2,

...,

n

[Ci.i(t)Ci~l.~(t)(£UAt)

[{[1 --Ci.j(t)]Ci+l.k(t

X I] + ) + C i . j ( t ) [1 - -

(A1.14)

[C,.i(t)£(1 - - U A t ) ] ,

The expected number of key particles in cell k in the (i + 1)th section resulting from key-non-key particIe collisions during At, p r o v i d e d t h a t t h e c o a l e s c e n c e is w i t h c e l l k i n t h e ( i + 1 ) t h s e c t i o n , is

=

(Al.13}

el+ 1.k(t)] }~ (~UAt)] +

2 .... ,m

{ [1 -- cg.j(t)]

(AI.12)

2,...,m

--C~.j(t)]Ci+1.k(t) +

Ci.i(t)[1 --C~+x.k(t)]}e(£Uht),

i

(AI.II)

i=1,2, ...,n--I

=l, 2,...,n--I

j,k=l,

(12UAt)]/£

= 1 -- U A t

) +

Ci./(t) [1 - - C i + , . ~ ( t ) ] } ( £ U A t ) ,

i

--

(A1.7)

The expected number of key-non-key particle collisions during At, provided that cell j in the ith section coalesces with cell k in t h e ( i + 1 ) t h s e c t i o n , is

{[1

[£

C i . j ( t ) ] Ci + ~.~(t) + C i . j ( t ) [ 1 - - C i + 1 . , ( t ) ] ,

j , k = I, 2 . . . . , m

i

t h e c e l l is s e l e c t e d a s o n e c e l l o f a c o a l e s c i n g p a i r , is

--

[1 --

(AI.IO)

1

j , / z = 1 , 2 , ..., m Collisions between non-key particles contribute nothing to the concentration of key

particles. T h e n u m b e r fraction of particles in a cell which d o not collide during At, provided that

i

=1,2,...,n--1

X k = ! , 2 , ..., m

The total expected number of key particles in cell k in the (i + 1)th section at t + At, p r o v i d e d t h a t t h e c o a l e s c e n c e is w i t h c e l l j i n t h e i t h s e c t i o n , is [Ci.i(t)Ci+l.k(t)(£UAt)

[{

[1 -- C~.s(t)]

Ci+l,k(t)l

X 1] +

C~+ ~.k(t) + C~a(t)

}(£U&t)(1

--e)]

[Ci+ z.k(t) £ ( I - - U A t ) ] , i

= 1, 2, ..., n - - I

Xk=l,

2..... m

[1 --

+

(A1.15)

The expected concentration of key part.icIes iq cell j in the ith section at r + Pt, provided that the coalescence is with cell k in the (i f 1)th section, is

--c,.j(~)lG*,.,(o

(t) +I11

IG.i(0G+,.,

c,.i(t)ll --ci+I.~(~)l~Q1 uat + c,_j(t)(l - r/M), i

= 1, 2. ___, n -

+

x (A1.16)

1

j,k = 1, 2, ___, m which is eqn. (I). The expected concentration of key particles in cell k in the (i + 1)th section at t + At, provided that the coalescence is with cell j in the ith section, is (t) +

IG.i(Wi+1.iz

C[l -

(r)]lx(l-

G.j(t)cl--i+l.k Ci,l.k

i

(t)(l

c,i(t)lci+,

-

ULU),

= 1, 2, ___, n -

k r f o! )]

,::

1

t-

(A1.17)

1

OF EQNS. (1) -4ND (5)

The probability that cell j in the first section is seIected as the first cell of a coalescing pair is 1

(A2.1)

nXn2

The conditional probability that cell k in the second section is selected as the second cell of the coalescing pair, provided that cell j in the first section is seIected as the first cell of the coalescing pair, is. 1 m The probability that cell j in the fit section is selected as the fiit cell of the coalescing pair and cell k in the second section is selected as the second cell of the coalescing pair is 1 nXm

Xl

m

(M-4)

nXm

The conditional probability that either cell k in the (i f 1)th section or cell k in the (i - 1)th section is selected as the second cell of the coalescing pair, provided that cell j in the ith section is selected as the first cell of the coalescing pair, is 1

(A2_5)

2Xm

The probability that cell j in the ith section is selected as the first ccl: of the coalescing pair and cell k in the (i + 1)th (or the (i - 1)th) section is selected as the second cell of the coalescing pair is 1 2Xm

(A2.6)

which is eqn. (5).

which is eqn. (2).

2_ DERIVATION

1

1 -xn X m

j, k = i, 2, ___, m

M’PENDIX

as the two cells of the coalescing pair can be determined in an identical manner. For an intermediate section, the probability that cell j in the ith section is selected as the first cell of a coalescing pair is

(A2.3)

which is eqn. (4). The probability that celli in the nth section and ceU .EE in the (n - 1)th section are selected

REFERENCES

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231 10

R. L. Cud, Dispersed phase miring: 1. Theory effects in simple reactors, AIChE J., 9 (1963) 175.

11

L. A_ Spielman

12

13 14

and O_ Levenspiel.

A Monte

and 15

Carlo

treatment for reacting and coalescing dispersed phase systems, Chem_ Eng. Sci.. 20 (1965) 247. A. Kattan and R. J. Alder, A stochastic mixing model for homogeneous, turbulent, tubular reactor, AIChE J., 13 (1967) 580. I. Komasawa. E. Kunugita and T. Otake, Collision model for dispersed phase in packed liquid-liquid beds, Kagaku Kogaku, 30 (1966) 37,237_ M. A. Zeitlin and L. L. ‘I’avlariies, FIuid-fluid interactions and hydrodynamics in agitated disper-

16

17 18

sions: a simulated model, Can. J. Chem. Eng-, 50 (1972) 207. G. K Patterson, Average molecular weight distributions in stirred-tank reactors by random coalescence dispersion simulation, presented at the 79th AIChE National Meeting, Houston, Texas, March 16 - 20.1975. H. Gelves-Arocha, Mixing and segregation of particulate sofids in a motionless mbcer. Master Thesis, Kansas State Univ., 1973. H_ A. Meyer, Symposium on Monte Carlo Methods, Wiley, New York, 1956. Y_ A. Shreider. The Monte Carlo Method. Perganon

Press,

New

York,

1966.