Stratification by size in particulate fluidization and in hindered settling

Shorter communications C -

co

=

by transformation (7), provided that the diffusivities of the species under consideration with respect to the main solution are equal. The application of the results to specific problems of interest are straightforward. The problem of turbulent mixing with first-order reactions is discussed elsewhere [4].

1 - Cf&)exp(-nt) n

Therefore, from (7) and (24), we obtain the solutions the reacting species as:

for

(25)

Acknowledgement--The author wishes to thank Dr. M. E. GRAHAM and Dr. J. D. MCCLURE for reading the manuscript. YIH-HO PAO

In the foregoing analysis, we have shown that, with the given initial and boundary conditions (4) N (6), and with properly determined &(t), the problem of unsteady mass transfer with any type of first-order reactions can be reduced to the corresponding problem with no chemical reaction

Flight Sciences Laboratory, Boeing Scientific Research Laboratories, P.O. Box 3981 Seattle, Washington, U.S.A.

REFERENCES [l] DANCKWERT~P. V., Trans. Faraday Sot. 1951 47 1014. [2] CARSLAWH. S. and JAU~ERJ. C., Conduction of Heat in Solids, 2nd Ed. Oxford University Press 1957. [3] CRANK J., The Mathematics of Difision, Oxford University Press 1956. [4] PAO, YIH-HO, AZAA Journal 1964 2. In press.

Chemical Engineering

Science, 1964, Vol. 14, pp. 696-700. Pergamon Press Ltd., Oxford.

Stratification

by size in particulate fluidization

Printed in Great Britain.

and in hindered settling

(Received 12 September 1963; in revised form 23 March 1964) THE PURPOSEof the present communication is to point out certain incorrect statements which have been made with respect to classification by size in liquid-fluidized beds and in hindered settling, and to propose a simple formula by which sizing in particulately fluidized beds can be rationally

analysed. Some remarks are also made on a critical difference between particulate fluidization and hindered settling in this respect. The expansion of a particulately fluidized bed of uniformly sized particles is most conveniently described [l] by an

0.6

0.4i

I

20

I

40

I

60

FIG. 1. Representation

I

60

I

100

I

120

I

140

I

160

de p of equation (3) for laminar flow.

696

I

160

I

200

Shorter communications

I

I

- 60001

I

/

i 4000

/

!,'/‘/

I

A/

!O

60

40

FIG. 2. Representation

60

120

100

160

160

140

of equation (3) for laminar flow (solid lines) and of equation (5) (dashed lines). bouring grain centres, and that their mutual orientation remains unchanged during bed expansion, then

equation of the form V = vzc = Vo@

(1) 1 -

and ZAKI [2] have shown that if the ratio of particle to tube diameter is sufficiently small, then VO is the terminal free settling velocity of the individual particles, equation (1) is applicable to sedimentation of the same particles and, for a given particle shape, the index n depends only on the free settling Reynolds number. The conditions of negligible particle-to-tube diameter ratio and constant particle shape will be assumed in the subsequent discussion. For spheres in laminar flow, where inertial forces are negligible, n in equation (1) has been found [2, 31 to be 4.65 and Vo is the Stokes terminal velocity given by

6

=

(1 -

ernf) d3/D3

(44

FUCHARDSON

v.

=

(PP -

P)

sd2

Equations formula v

0

(PP- P) gd

C(PP -

pv

pm-1,/z pnl

6

140

I.

d

(2b)

044 p

(2a) and (2b) can be represented =

::ioo

120

For spheres in the Newton turbulent region, on the other hand, n has been found [2] to be 2.39 and VO is the terminal velocity which is well approximated by 4,3

f 16C

(2a)

18P

voz =

IEIC

100

80

by the single 60

gl’m dWm)im

z

kd(3-m)/m

(2)

where m varies from 1 in the Stokes region to 2 in the Newton region and, within a given fluid regime, k is a constantfor particulate fluidization of constant density spheres in a given liquid at a fixed temperature. Combination of equations (1) and (2) gives V = kd(s-m)lm cn (3)

I 40

2c

C

Plots of equation (3) for m = 1 and n = 465 are presented in Figs 1 and 2. Assuming, as does JOTTRANLI [l], that particulately fluidized spheres are uniformly spaced at a distance D between neigh-

40

60

60

100

I20

d, P FIG. 3. Representation of equation (5a) for laminar flow (solid lines) and of equation (4a) (dashed lines).

697

140

Shorter communications For uniform-size spheres crnf averages about 0.465 [4], and this value is used to represent equation (4a) as a family of straight lines on Fig. 3. Combination of equations (3) and (4a) to eliminate E leads to

The tendency to segregation during particulate fluidization of a binary solid mixture is best analysed by focusing on its driving force-the difference in bulk density of the individual beds [Sa, 8b]. The bulk density of a particle-fluid mixture is given by pe = (1 - E) pP + tp = (1 - 4 (pz,- PI + p

A plot of equation (5a) for m = 1, taking n = 4.65 and cm, 1 0.465, is-shown in Fig. 3. A sir&lar plot (with n = 4.0 and crnf = 040) is Dresented by JOTTRAND111.who maintains that the mini&a in the curve-s represent a critical value of bed expansion below which constant density spheres will no longer classify by size, the smaller above the larger, and above which it is no longer possible to have a homogeneous bed with a precise upper limit. The magnitude of this socalled critical value can be obtained analytically by differentiating equation (5a) with respect to d, taking k and V as constant and D as the dependent variable, and setting the derivative of D equal to zero. The result of this operation is

PBL

-

PBS

=

@P - p) (CS- EL)

(74

If both beds occupy the same constant diameter column, in which they are supported by a common fluidizing stream, then they are being subjected to the same superficial velocity V, and therefore by equation (l), VOLELn

=

v&s ES”

(14

Equation (la) assumes that both particle sizes are either in the Stokes region or in the Newton region, rather than one or both in the intermediate region, in which case n would be different for each size. The latter eventuality would complicate the subsequent equations but would not invalidate the basic analysis. Substituting for the free settling velocities in equation (la) according to equation (2),

Therefore, by equation (4a), the critical porosity cC is given by 3mn (6) Cc = 3 - m + 3mn In the Stokes’ law region (m = 1, n = 4.65) this gives cc = O-8746, as can be seen also in Fig. 3, and in the region of Newton’s law (m = 2, n = 2.39) cc = 0.935. In other words, according to JOIXXAND’Sanalysis, classification by size will not occur in a liquid-fluidized bed until the bed has expanded beyond these porosities. Aside from the fact that experimental data [2, 5, 61 on sizing in liquid-fluidized beds show a continuous tendency to stratification from the minimum fluidization point to the point of total elutriation, this analysis is also contradicted by reference to Fig. 1. Here it is seen that for particulate tluidization of two different sizes of particles of the same density by the same fluid stream, the porosity associated with the smaller size is always greater than that associated with the larger. Unlike Fig. 3, no minima are shown by these curves. This means that the bulk density of the smaller particles during particulate fluidization is always less than that of the larger particles, an effect which leads to the equilibration of the lower density bed of smaller particles above the higher density bed of larger particles [7, 81. JOTTRAND'S contrary interpretation of Fig. 3 arises largely out of the misleading nature of the minima noted therein. These minima disappear as soon as either clearance distance D-d or void fraction E (as in Fig. l), rather than centre-to-centre distance D, is plotted against d. The confusion is reinforced by inconsistently treating V as interstitial rather than as superficial velocity. JOTTRAND[l] also makes the assertion, on experimental grounds, that for high concentrations of two powders of equal density but different size, no classification occurs during water fluidization unless the size difference is sufficiently great that there is at least a twofold difference in free settling velocities. Referring to equation (2), this statement implies that minimum sphere diameter ratios of 212 :l (for m = 1) to 4 : 1 (for m = 2) are required for stratification to occur during dense-phase particulate fluidization. The necessity of such large diameter ratios is contradicted by both laminar and turbulent liquid fluidization data [6, 7, S].

(7)

The difference in bulk density between a bed of large particles and a bed of small particles of equal density is therefore

ES -= EL

Eliminating PBL -

& (I-m)/mn ( ds )

(8)

ES in equation (7a) by means of equation (8), PBS

=


,d CL

If there were no forces opposing stratification by size in a fluidized bed, then the slightest positive value of the righthand side of equation (9) would signify such stratification. In fact, however, factors opposing stratification, such as particle circulation [9] and hydrodynamic instability [IO], are inherent in the operation, even for smooth particulate fluidization. The minimum difference in bulk density necessary to promotemeasurablestratificationbysizeundervarious circumstances is therefore subject to experimental determination [8b]. Nevertheless, one can generalize to the effect that the greater the value of the bulk density difference as predicted by equation (9), the greater the tendency towards segregation by size. Equation (9) therefore predicts that, as in free settling classification, sizing occurs more readily the higher the values of dL/ds and (pP - p)‘, and the lower the value of m (the effect of decreasing which on the index (3 - m)/mn is always greater than the reverse effect of the corresponding increase in n). In addition, this equation also predicts that the stratification tendency is enhanced by increasing EL and hence ES [see equation (S)], that is, by expanding the particulately fluidized bed. It should be noted that this effect is a continuous one, however, and there is no reason to suppose that contracting the bed will eliminate rather than merely reduce stratification, unless the other factors on the right-hand side of equation (9) are sufficiently * A recently completed experimental study [8b], however, points tolittleeffectof@, - p) on stratification by size and to the possibIe applicability of the reduced bulkdensity difference ratio (pBL - pBS)/(pp - p) in equation (9) as the appropriate index of stratification.

698

Shorter communications small as to result in the bulk density difference being reduced below the minimum necessary to promote stratification. Sedimentation of suspensions containing particles of mixed sizes differs from fluidization of the same particles in that the superficial velocity of the fluid relative to the particles is different for each particle size in sedimentation but not in fluidization, assuming a chamber of fixed cross-section in both cases. Thus while equation (1) applies to each layer of a particulately fluidized bed once stratification is complete, with the same value of V for each layer, it would only start to apply in sedimentation if the settling chamber were sufficiently deep to bring about complete segregation of each particle size in the settling zone (as opposed to the compression zone at the bottom of the chamber, where equation (1) is completely inapplicable). Even for this experimentally improbable situation, however, a different V would apply to each particle size, except under special conditions discussed below. The additional degree of freedom associated with each particle size in sedimentation is important when coupled with the fact that, under certain circumstances, a swarm of large particles can actually settle at a lower velocity relative to the walls of the settling chamber than a swarm of smaller particles. The analogous effect in fluidization has no bearing on classification. In hindered settling this effect arises out of the increasing importance of back-flow as particle diameter is increased, for a suspension containing a fixed number, N, of uniform-size spheres per unit volume. The concentration of such a suspension is given by 1 - E = (7r/6) (d3N>

(4)

This equation is a more general relationship than equation (4a), the latter being limited by JOTTRAND’Sassumption of unchanging mutual orientation of the spheres. Elimination of E in equation (3) by means of equation (4) results in V = kd(3-m)lm [l - (n/6) (Nd3)ln

(5)

which again is more general than equation @a). Equation (5) is plotted in Fig. 2 for several values of N (number of particles per cubic micron). Differentiating equation (5) with respect to d, taking k and N as constant and V as the dependent variable, and setting the derivative of V equal to zero, yields (7r/6) (Nd3) =

3mn

eM = 3 - m + 3mn

(10)

That this porosity is the same as the one given by equation (6), which was derived with D rather than Vas the dependent variable, should not conceal the fact that the physical constraints applied were different in the two derivations.? The result for equation (7) with m = 1, n = 4.65 is shown in Fig. 2 as the locus of the maxima on the lines of constant N. One implication of this result is that especially in the lower right-hand portion of Fig. 2, a swarm of small spheres released above a settling swarm of larger ones of the same density can overtake the larger ones if N for the larger ones is greater than, equal to, or not much less than N for the small ones. On overtaking the larger spheres, the smaller ones will be slowed down to the same velocity as the large ones. Such velocity equalization of artificially classified particles will most readily occur at high concentrations. Conversely, different velocities for each particle size in a mixedsize suspension will tend to occur below these concentrations. That these limiting concentrations are very much greater than is usually appreciated is confirmed by the recent experimental work of KAYE and DAVIES [ll]. This work also negates the blanket claim by COULMN and RICHARDSON[12] that “if the range of particle size is not more than about 6 : 1, a concentrated suspension settles with a sharp interface and all the particles fall at the same velocity’.‘, especially considering that in the context of their use of the term, “concentrated” refers to that wide span of porosity (E N 0.5-0.99) within which particle interactions preclude free settling. The work of KAYEand DAVIES[I l] also shows that a sharp interface between supernatant liquid and particle swarm during sedimentation does not necessarily signify the absence of stratification by size within the swarm. The previously cited works [2, 5-81 on particulate fluidization similarly show that the presence of a precise upper limit to a fluidized bed does not necessarily mean that the particles within the bed are not stratified by size. Acknowledgement-The work out of which the present communication arose is being supported by a grant from the National Research Council of Canada.

NOTA~ON

3-m 3 - m + 3mn

c

Therefore, by equation (4), the porosity EM at which increasing particle diameter alone gives rise to a maximum settling velocity is

d D g k

t It should be emphasized that equations (6’ and (7) are not specific to their respective antecedents, equations (Sa) and (5) respectively. Thus equation (7) could just as well have been obtained from equation (5a), taking D constant and V as the dependent variable, or equation (6) from equation (5) with- V constant and N as the independent variable. Nor should these critical porosities be confused with the porosities at which the mass sedimentation flux pp V (1 - l) is a maximum, namely, E = n/(n + 1) for constant d but variable N, and E = 3mn/(3 + 2m + 3mn) for constant N but variable d.

m 1(: V

V6 vo E

699

Dimensionless coefficient in equation (2) which varies with fluid regime Particle diameter Centre-to-centre distance between neighbouring particles Acceleration of gravity Dimensional coefficient in equation (2) which varies with fluid regime, fluid properties and particle density Index of fluid regime in equation (2) Richardson-Zaki index in equation (1) Number of particles per unit volume Superficial velocity of fluid relative to particles; actual velocity of fluid relative to wall above fluidized bed; actual velocity of mono-size particles relative to wall in sedimentation Interstitial velocity of fluid relative to particles Terminal free settling velocity Fractional void volume; porosity

Shorter communications Porosity at which D is minimized by varying only d at constant V Porosity at which V is maximized by varying only d EM at constant N Porosity at minimum fluidization point Emf CL Viscosity of fluid P Density of fluid Pa Bulk density PP Particle density

Subscripts L Large S Small B. B. PRUDEN N. EPSTEIN Chemical Engineering Department University of British Columbia Vancouver, B.C.

REFERENCES

HI

PI 131

[41 PI b51 [71 PI Ml WI 191 DOI Hll WI

JO~RAND R., Chem. Engng. Sci. 1954 3 12. RICHARDSONJ. F. and ZAKI W. N., Trans. Znstn. Chem. Engrs 1954 32 35. RICHARDSONJ. F. and MEIKLER. A., Trans Znstn. Chem. Engrs 1961 39 348. EPSTEINN., D. Eng. Sci. Thesis, New York University, 1953. BR&Z W., Chem. Zng. Tech. 1952 24 60. ANDRIEUR., Ph.D. Thesis, University of Nancy, France, 1956. VERSCH~~RH., Appl. Sci. Res. 1950 A2 155. LEVA M., Fluidization, p. 92. McGraw-Hill, New York 1959. LE CLAIR B. P., Two Component Fluidization, M.A.Sc. Thesis, U.B.C., 1964. PRUDENB. B., Particle Size Segregation in Particulately Fluidized Beds, M.A.Sc. Thesis, U.B.C., 1964. HAPPEL J. and BRENNERH., Amer. Inst. Chem. Engrs J. 1957 3 506. JACKSONR., Trans. Inst. Chem. Engrs 1963 41 13. KAYE B. H. and DAVIES R., Symposium on the Interaction between Fluids and Particles, London. The Institution of Chemical Engineers, 1962, p. 22. COULSONJ. M. and RICHARDSONJ. F., Chemical Engineering 2, p. 511. Pergamon Press, London 1955.

700