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The stability of slugs in fluidised beds of fine particles
THE STABILITY OF SLUGS IN FLUIDISED OF FINE PARTICLES A THEORY
BASED
ON PARTICLE
PICK-UP
FROM
THE
BEDS
WAKE
J. R. F. GUEDES DE CARVALHO Departamento de Engenharia Quimica. Faculdade de Eugenharia, Rua dos Bragas. Porto, Portugal (Received 1 August W30: accepted 22 lunlrary 1981) Abstract-A theory for slug stability in fluidised beds of fine particles is developed based on the idea of gas circulation inside bubbles. Data on the intensity of circulation inside gas slugs in liquids are reported and an analogy is drawn with slugs in fluidised beds; this information is coupled with data on gas pick-up velocity for flow over granular beds, to yield a simple criterion for slug stability. A test of this criterion is provided by comparison with data on the stabiity of freely slugging beds. An X-ray study of slugs in fluidised beds at various pressures is also described that gives further support IO the predictions
of the theory.
INTRODUCTION
It is well known that while most liquid-solid
systems fluidise homogeneously, the gas fluidisation of solids is usually aggregative with large gas bubbles passing through the bed. This behaviour is generally attributed to the low viscosity of gases and the large density difference between gas and solids. This idea gained some experimental support from the fluidisation of light solids with gases under pressure, which proved to be of the particulate type[l], and from the fluidisation of heavy solids with water, which proved to be aggregative[2]. If to the fluidising water was added some glycerol. thus increasing the viscosity, the state of uniform fiuidisation could be reached again, showing the effect of viscosity. Various workers tried to explain the origin of bubbles in fluidised beds. Rice and Wilhelm[3] have shown that the lower surface of a ffuidised bed supported merely by the flowing gas is unstable. Their analysis shows that a perturbation on the lower surface of a bed tends to grow, and that this process is a function of the effective density and apparent viscosity of the dense phase. Others[4-61 have made similar analyses and the general conclusion is that any small perturbation about the state of uniform fluidisation is initially magnified, giving rise to bubbles. According to these authors, the difference between gas and liquid fluidised beds arises from a much lower initial rate of growth of perturbations in the latter. Molerus[7] distinguishes between stable systems (liquid-solid type) and unstable systems (gas-solid type). However his conclusions, like those of the other authors, are of a qualitative nature. More recently Verloop and Hertjes[S] developed a criterion for the transition from homogeneous to heterogeneous fluidisation based on a comparison between the rise velocity of a porosity fluctuation and the longitudinal propagation of an equilibrium disturbance. However, their predictions are not always correct; in particular the influence of gas pressure as described by Leung[l], is not predicted.
Harrison et al. [2] have looked at the different types of fluidised behaviour in another way. For any given system they imagined a bubble rising through the bed and then proceeded to consider the stability of this bubble (a bubble in a liquid-solid system being a “liquid bubble”). The development of this idea has led to the concept of maximum stable bubble size. Systems in which the diameter of the largest stable bubble is comparable to the particle size fiuidise particulately and beds in which the largest stable bubbles are of a much larger size than the particles fluidise aggregatively. This theory provides a distinction between homogeneous and heterogeneous fluidisation in general agreement with experiment [9], and at the same time predicts the existence of a maximum size of stable bubble in systems in which bubbles form. An alternative theory which might explain a limitation in the size which bubbles can reach, is the theory of bubble breakup based on Taylor instability of the bubble roofflO-121. The application of this theory to the stability of fluidised beds of fine powders under pressure is considered in detail elsewhere [13]. TAETAEORYOFAARRtSONetsLANDlTS EXTENSIONTO SLUG STABILITY
Harrison et uf.[2] postulate the existence of gas circulation inside bubbles rising in fluidised beds, by analogy with gas-liquid and liquid-liquid systems; the gas inside the bubble is dragged down by the particulate phase (Fig. 5) and flows up in the middle. The maximum upwards velocity along the axis of the bubble is estimated to be UC= Lr, = 0.7bJ(gD.)
where iJb is the bubble rise velocity, g the acceleration of gravity and 0, the equivalent diameter to the bubble; the velocity UC is measured relative to axis fixed with the bubble. A solid particle is then imagined to be transferred from the wake to the inside of the bubble and
I349 CES Vol. 36. No. 8-F
(1)
1350
J. R. F.
GUEDES DE CARVALHO
consideration is-given to its possible behaviour: (i) if the upwards velocity, U,, is greater than the terminal free settling velocity of the particle, U,, the bubble tends to get filled with particles from the wake and the bubble is not stable; (ii) if iJ, < U, the particle falls back on the wake and the bubble is stable. For a given fluid particle system at given pressure and temperature, V, is fixed and because U, increases with D,, a bubble size exists for which U, = U, and this is the maximum size for a stable bubble in the system. In beds with a high ratio of height over diameter, H/D, slugging is often observed and the above theory on bubble stability may be adapted to the determination of slug stability. Experiments described in this paper suggest that the maximum upwards velocity in the slug, V,, is related to its rise velocity, U,, by U, = 0.9l_J, = 0.315d(gD)
(2)
Fig. 1. Criterion for bubble and slug stability based on of Harrison
Gas pressure in bar is indicated on the figure next to each point
cm9 gas
where D is the bed diameter. By analogy with bubble stability it may be said that slugging is not possible if CJ, < 0.315~(gD).
(3)
If a Reynolds number under free settling conditions is defined for a particle with diameter d, Re, =(&d/p) where p is the fluid density and CLits viscosity, it may be shown to depend on Ga only[l4]. Re, = f(Ga)
(4)
where GO = p(p, - p)gd3/p*, p. being the particle density. For spherical particles eqn (4) is given approximately by the following expressions (14) (in the form Gu = f-‘(Re,)) Ga = lgRe, Ga=18Re Ga =; Re:
(Ga < 3.6)
t +27Re’ . .687 I
(4.1) (3.6
(Gu >Q I@).
From(4) the condition thus be written as
(4.2) (4.3)
for bubble (or slug) stability may
UC < Wlpd)f(Gu).
(5)
Substitution of U, from (1) and (2) yields, after some algebraic manipulation
for bubble stability, and
for slug stability. Figure 1 shows these two conditions and it may be used to determine the maximum stable bubble size for a
the theory
et al. [2].
0.95x103 48 0.95x 103 82 2.90x I@ 64 2.9OxlV 64 2.92x I@ 68 2.69x t# 74 290x103 211
0.05 0.05 0.10 0.10 0.05 0.10 0.10
N2 N2
2,
Stable slugging
Unstable slugging
A E
$
c%* is N2
0
-o-
given fluid-particle system, by determining the relevant value of Ga and reading the corresponding ordinate off the bubble stability line. On the other hand, for a given gas-solid system in a tube both Go and the left band side in (5.2) may be determined, and the corresponding point plotted in the graph; slugging is predicted to be unstable for points above the solid line and stable for points below that line. In the extreme cases of free fall under laminar flow (Ga < 3.6) or fully turbulent flow (Ga > lo’), inequalities (5.1) and (5.2) lead to very simple analytical expressions, upon substitution of J(Go) by (4.1) and (4.2) respectively. Comparison of theory with experiment requires a clear understanding of the meaning of the stability of the slug. If the coalescence of small bubbles leads to the formation of a slug that is unstable, the foregoing theory predicts that it will break up by filling in from the lower surface. Evidence for slug instability in Ruidised beds may be sought from two sources; X-ray “viewing” of the bed (to be discussed later in this paper) and observation of bed surface fluctuation. In freely slugging beds the ratio of slug length to bed diameter increases with gas flowrate and height above distributor[M]; if slugs are unstable it is expected that the amplitude of surface fluctuations is only a fraction of D, even if the gas flowrate is high and the ratio H/D is large. This was the reasoning behind the method used by Guedes de Carvalho[l6] in order to study the influence of gas pressure on the stability of slugs. Detailed reports of the experimental findings are given elsewhere [ 17, 181 and the data relating to the stability of slugs are represented in Fig. 1. It may be seen that while the correct behaviour (stability of slugging) is predicted for glass particles
The stability of slugs in fluidised beds of fine particles
1351
211 pm in diameter fluidised by air, for very fine particles the impossibility of slugging is predicted, in contradiction with experimental observation[l7]. An explanation for this “misbehaviour” of beds of fine particles (< 100 Cm), based on the relative difficulty of pick up of particles from the wake has been suggested[19]; this is discussed in detail in the next section.
THESTABILITYOF THE LOWER SURFACE OF sLucs
Harrison
et ob[2] looked at the forces exerted by circulating gas on particles placed near the “center” of a bubble, but no consideration was given to particle transport from the bubble surface to this central position. A possible mechanism is that particles are picked up and entrained into the bubble when the drag exerted by the circulating gas over the lower surface of the bubble exceeds a certain value [ 191. Bagnold[20] studied the forces on a particle at the surface of a bed of sand across which air blows horizontally, as in a desert; the strength of the wind being measured by the drag velocity u* = g(~/p) where Q is the magnitude of the shear stress exerted by the wind on the surface. Bagnold found that there is a “threshold” drag velocity, (u,),, above which the horizontal drag on the particle is enough to overcome the opposed moment due to gravity, thereby causing the particle to ride over the neighbour against which it rests. This analysis may be applied to the stability of a slug (or bubble) in a fluidised bed, the surface of the desert in Bagnold’s analysis being equivalent to the lower surface of the slug and the wind over this surface being due to the vortex within the bubble. It should be stressed that the situation for which Bagnold’s analysis was developed does not match exactly the conditions over the lower surface of slugs. (i) Bagnold considered unidirectional gas movement over a sand bed and that is not the case inside the slug; however drag velocity is the important parameter and this can be calculated approximately for the flow conditions inside the slug. (ii) Gas percolation through the slug wake can be expected to help lift the particles; this may be true to some extent, but the force on a particle decreases markedly as it is lifted up from the bed on which it rests. Clift et ol.[21] applied Bagnold’s analysis to the analogous problem of pick-up of particles by a gas jet issuing into a fluidised bed and found good agreement between theory and experiment.
O.O11o
I 1OD
d@ml ’ lo
Fig. 2. Dependence of threshold drag velocity on particle size for sand in air (at room temperature), from eqn (6).
mechanism of particle pick-up, Bagnold [20] suggested expression for the threshold velocity (~1, = AX&$. - p)gdlp)
(6)
“constant”. Yalin[22] used where A is a dimensionless dimensional analysis to show that A depends only on (Re,), = p(u,),d/p. The relation between A and (Re*), has to be determined experimentally. Figure 3 shows most of the experimental data available, as given by Tnversen et af.[23] in a study concerned with aeolian processes on Mars. In that study a wide range of particle sizes and densities was covered and it may be seen that for (Re,), > 5 the value of A is constant al about 0.12. For the very fine particles (i.e. low (Re,),), interparticle forces may be important and it is possible that A is not a function of (Re*), alone[23]. Also shown in Fig. 3 are the data of Zenz[24] on saltation velocity, even though it refers to the slightly different situation in which single particles are picked up from a smooth horizontal surface. The values of (u*), for this case were computed from Blasius equation for turbulent flow in smooth pipes. The plot in Fig. 3 has the disadvantage of in&ding (u*), in both dimensionless groups and a more convenient plot of A vs Go is shown in Fig. 4 where only values corresponding to the solid line in Fig. 3 are reproduced. Values of (u,), for a given gas-solid system may be easily found from this plot.
Analysis of the “threshold velocity” The relation between threshold velocity
and particle diameter is illustrated for sand particles in Fig. 2. For large particles the drag velocity decreases with the square root of particle size until a diameter is reached (80 pm at 1 atm) below which the particles are totally immersed in the laminar sub-layer and increased threshold velocities are necessary to pick them up. This feature of very small particles could explain the failure of the theory of Harrison et a/.[21 to predict the behaviour of beds of fine powders. Based on a
an
a2
0.t
Fig. 3. Variation
of A with threshold
drag Reynolds
number.
1352
I
J. R. F. GUEDES DECARVALHO
. . . . ....4
0.1
1
.
.
. * . . ..
I
10
.
* . . . . .. 1
xl0
.
.
I.
i,doo &
plified analysis of the shear stress over the lower surface of the slug. The flow near the rim (Fig. 5) may be represented (as an approximation) by plane flow towards a stagnation point and this leads to a simple analytical solution for the drag velocity; the two dimensional simplification seems reasonable as the region where the drag velocity is highest is far from the axis of the slug. For plane flow towards a stagnation point the stream function is tj = kxy and the velocity parallel to the plane, ~O,L!OO u, varies with distance from the stagnation point, x, as
Fig. 4. Dependence of A on Galileo number.
u=kr
The plots in Figs. 3 and 4 are not applicable for liquid-solid systems as it has been reported[20] that for such systems the relation between A and (Rc,), is significantly different. At high (Re,),, the value of A for liquids is approximately double that for gases.
where k is a constant. The solution for the drag velocity over the surface is[261
Flow over the lower surface of a slug The analysis of slug stability based on particle pick-up from the wake requires the determination of the drag velocity over the bottom of the slug. If (I)*),,,*~ represents the maximum value of the drag velocity over the lower surface of the slug, particle pick-up and therefore slug instability should occur for (v*), <(l)+Jmax. For beds of fine particles the gas “throughflow” in the slug is negligible and an estimate of (u*),,,~~ may be obtained from data on circulation inside slugs in liquids. The two situations are indeed analogous. The analysis of flow around a slug[lS] shows that the particle velocity on the slug surface, I+,, is equal to the liquid velocity in the corresponding gas-liquid interface. Now the tangential component of the gas velocity v, (Fig. 5) has to be equal to u, otherwise a non zero pressure gradient apI& would show over the slug surface. The gas velocity perpendicular to the surface, u,,, is of order Umf and therefore negligible for systems of fine particles. As a result uplrtiFlc= vsnS over the slug surface and this is a “no slip” condition, similar to that observed on the gas-liquid interface. Filla et a!.[251 used ammonium chloride to help visualize the circulatory streamlines inside air slugs in water; a plot of the streamlines for a slug with length equal to twice the tube diameter was also obtained, based on a numerical solution of the equation of inviscid flow inside the slug. The shape of the streamlines suggests a sim-
u* = d(rdp)=
l.llu(Rex)-“4
(71
63)
where Re, = pux/p. The numerical solution for flow inside a slug does not give absolute values of velocity, but relative velocities between any two points may be calculated from it. In Fig. 6 the horizontal velocity close to the lower surface of the slug is given, relative to the maximum upwards centre line velocity, UC, as a function of distance from the rim, X. The velocity increases initially almost linearly with distance, reaches a maximum, and then decreases again as the axis is approached. The analogy with flow towards a stagnation point clearly ceases to apply at some distance from the rim. Figure 6 shows that the approximation is reasonable until x/D = 0.1, with ul U, = 7.4(x/D) at that point. If this is taken to be the point where the drag velocity is a maximum, substitution in eqn (8) gives (v*),.,
= 0.821 U,(l3.5,u/pDU,)“’
(9)
and all that is needed is a good estimate of UC. Filla et nl.[25] described a technique in which a slug is kept stationary inside an inverted rotameter tube by means of a downtlow of water, and a hot wire anemometer is used to measure centre line velocities inside the slug. In the present study, this technique was used to
X
Fig. 5. Idealised flow pattern near rim of slug.
Fii. 6. Dependence of horizontal velocity component on distance from rim for ideal flow inside slug &ID = 21.
The stability of slugs in fluidised beds of fine particles
is proportional to slug rise velocity the above value UC = 0.22 m/s suggests U, = 0.9U.. If this relation is assumed to hold for tube diameters other than that used in the experiment described, then
measure velocities inside slugs of various lengths, L,, held in a tube with average diameter D = 0.05 m. The experimental velocities are shown in Fig. 7 as a function of distance from the slug nose, z and the solid line corresponds to the theoretical values (i.e. numerical solution) with UC = 0.22 m/s. It may be seen that near the nose of the slug, theory and experiment are in good agreement; but as the bottom of the slug is approached a marked increase in gas velocity is observed. This is probably due to oscillation of liquid in the wake. In fluidised beds the dense phase is very “viscous” and such oscillations are not observed [27]. The velocity of rise of a slug in a tube of dia. D is U, = O.Xg(gD) and if the upwards circulation velocity
0.75
I
UC = 0.315d(gD) and substitution
0
= 0~~1)k18g318~lr4p-~~~~
(11)
Criterion for slug stabifity Equation (11) suggests that the maximum drag velocity over the bottom of a slug increases with tube diameter, although weakly, and since the threshold velocity for a given gas-solid system is fixed, there is a tube diameter for which (tl*),,, = (u,),, and slugs are not stable in beds with diameters larger than that. From (6) and (11) slugs are not stable if
a.rnn with
(10)
in (9) gives finally,
(u*),.,
c Cs-O.(6m 0 a.10 m o.vJm I
--Thy
1353
lJr=022mh
(Pa-;‘gdl”’
A
< 0.66 (F)II:
(12)
[
and this may be re-written
Fig. 7.
The possibility of slug instability increases with creasing pressure, whereas it is almost independent temperature as for most gases the product (pp) is sensitive to temperature changes, and is proportional pressure.
Gas circulation velocity along axis of “stationary” slug.
[
3
.
,,.I.’
s
.
10
.
. ..p
’
..,,*I
’
102
’
Id
’
.‘._
Ga
Fii. 8. Criterion for slug stability from theory in this paper Gas pressure in bar is indicated on the figure next to each point Stable Ref.
slugging
0.95 x 103 0.95x 1oJ 2.90x IO’
48 82 64
0.05 0.05 0.10
N, N, Nt
Ml ;;;;
n T
2.90x 2.92x 2.69x
64 68 74
0.10 0.05 0.10
co2 N, co2
tt:;
:
WI
0
LCP LO’ 10’
as
Unstable slugging
$ 0
inof into
J. R. F. GUEDES DECARVALHO
1354
Figure 8. shows a plot which may be used to determine the possibility of slug instability for a given gas-solid system in a tube. The full line corresponds to a plot of (A/0.66)’ vs Ga and as in Fig. 1, points above this line correspond to systems for which slugs are expected to be unstable, whereas for points below the line slugging should be stable: for points close to the line some degree of uncertainty is to be expected because of the approximations involved. For Ga > 3OMl the value of A is constant (A = 0.12) and slugs are stable if (13.1)
EXPERBIEWAL
EVIDENCE
Data on the behaviour of fluidised beds under pressure are scarce, but there is a widespread belief that fluidisation becomes “smoother” at higher pressures; “smoothness” being associated with the presence of smaller bubbles and therefore with smaller pressure fluctuations near the distributor. Bubbling bed behaviour Guedes de Carvalho and Harrison[l7] showed that fluidisation is not necessarily smoother at higher pressures, by Ruidising different powders in a narrow and long tube and observing their bubbling characteristics; al1 the powders were closely sized to avoid difficulty in the interpretation of the results. Synclyst particles (d = 48 pm) fluidised by nitrogen in a 0.05 m diameter tube could be made to slug at pressures up to 6 bar, but the bed level fluctuations decreased markedly in amplitude for higher pressures; this was taken as an indication that slugs were not stable at the higher pressures. The corresponding points are shown in Fig. 8 and they seem to agree with the criterion for slug stability presented above. The same authors fluidised glass hallotini (68 wrn) with nitrogen in the same tube and observed no change in bubbling characteristics at pressures up to 22 bar; at any pressure the bed could be made to slug easily if enough gas was supplied. In Fig. 8 it may be seen that this is in agreement with the theory. Another series of experiments [ IS] showed that 64 pm glass ballotini fluidised by carbon dioxide in a 0.1 m diameter bed showed unstable slugging for pressures above 20 bar, as predicted by the criterion for slug break-up. Figure 8 summarises the data available to the author on fluidisation of fine powders at high pressure and the agreement with theory of slug break-up seems to be very good especially if it is considered that the theory is based on a simplified physical model and no adjustable parameters were used to match theory and experiment. However, it would be desirable to have experimental data for other powders and at even higher gas densities. Bubble visualization An obvious way to test the theory would be to look into bubbles or slugs and see if particles were entrained from the wake. Subzwari et a[.[281 and Varadi and
Grace [29] photographed bubbles in two dimensional fluidised beds and observed that the dominant mechanism of bubble break-up was by indentation from the roof. However, this cannot be considered as refutation of the present theory because for the pressure range covered by those authors the present theory would predict large bubbles to be stable. 0n the other hand two dimensional beds are not the best tool to test the present theory because the walls are likely to interfere significantly with circulation inside the bubbles. An alternative method was used by Guedes de Carvalho [16] who used the X-ray facilities at University College. London, to film gas slugs in a 0.05 m cylindrical bed filled with catalyst. The X-ray beam and the tine camera were fixed at 0.8m above the distributor while the bed was fluidised by nitrogen: the gas flowrate was just enough for occasional slugs to appear and so avoiding frequent coalescence in the section viewed. Exposure times were 1/125Os per frame and sequences of five consecutive frames are shown in Figs. 9(a) and (b). Runs 1 and 2 were taken at 15 frames/s and Runs 3,4 and 5 were taken at 20 frames/s. The pressures and the characteristics of the particles used in the X-ray studies are given in Table 1. Runs 1, 3 and 4 were made at pressures below those required for slug instability, as predicted from the present theory, and the lower surface
Run 1
Run Z(a)
Run 2(b)
Fig. 9(a). X-Ray pictures of rising slugs.
i355
The stability of slugs in fluidised beds of fine particles
boundaries and shatter on occasions); the pressure at which this occlirs is predicted with good approximation by the theory developed in this paper. CONCLUSIONS
theory of Harrison et a/.[21 on bubble stability is shown to disagree with experimental results for fluidised beds of fine powders. An extension of that theory is considered based on the phenomenon of pick-up of particles from a granular bed and on an approximate description of circulatory gas flow inside a slug. The resulting quantitative criterion of slug stability is shown to be in good agreement with available experimental data. The same criterion defines a range of experimental conditions for which data should be sought in order to check further the theory in this paper. The
Acknowledge~enrs-Part of this work was done while the author was at the Department of Chemical Engineering, University of Cambridge. Financial support from Calouste Gulbenkian Foungratefully acknowledged. NOTATION
Run 3
“constant” in eqn (6) particle diameter or average particle diameter bed diameter equivalent diameter of bubble acceleration of gravity Galileo number bed height constant, eqn (7) slug length PU, d/cl P(U& d/p
Run 5
Run 4
Fig. 9(b). X-Ray pictures of rising slugs.
PUlfi
coordinate along slug surface velocity along wall in flow towards a stagnation point; horizontal velocity over lower bubble surface bubble rise velocity maximum upwards velocity along bubble axis, relative to axis fixed with the bubble slug rise velocity free settling velocity of particles drag velocity over surface maximum drag velocity
of the slugs may be seen to be reasonably flat. In the sequence corresponding to Run 4 in particular, an indentation develops from the roof and plunges into the wake, but leaves it unaffected. However. in Runs 2 and 5, the pressure is slightly higher than required for the slugs to be unstable according to the present theory, and the lack of definition of the slug boundaries (the wake included) is particularly noticeable. Although these pictures do not elucidate the exact mechanism of slug instability, they show nevertheless that above a certain pressure slugs become unstable, (i.e. have ill defined
Table I. Experimental conditions in X-ray experiments Run
Pressure
(bar) 1
1
2
(a>
14
2
(b)
14
Ps
(kg/m3 1
d (Inn)
950
75
950
75
950
75
3
1
950
59
4
6
950
5s
5
12
950
59
J. R. F. GIJEDES DECARVALHO
1356
threshold
drag velocity perpendicular to slug surface particle velocity over slug surface gas velocity parallel to slug surface distance from plane of symmetry in flow towards a stagnation point; horizontal distance from rim, Fig. 5 distance from wall in flow towards a stagnation point; distance from lower slug surface distance from slug nose fluid viscosity fluid density particle density shear stress stream function gas velocity
REFERENCES
Leung L. S., Ph.D. Thesis, University of Cambridge l%l.
;;; Harrison D., Davidson J. F. and de Kock J. W., Tmns. Inst. Chem. Engrs London 1961 39 202. [31 Rice W. I. and WiIhelm R. H., A.Z.Ch.E.Z. 1958 4 423. [41 Jackson R., Trans. Inst. Chem. Engrs London 1963 41 13. Murray J. D., J. Fluid Me&. I%5 21 465. :; Pigford . ^_ R. L. and Baron T., Ind. Engng Chem. Fund/s 1965 4 61. [71 Molerus, 0.. Proc. intern. Symp. Fiuidization (Edited by Drinkenburg A. A. H.) p. 134, Netherland University Press, Amsterdam 1967. 1. and Heertjes P. M., Chem. Engng Sci. 1970 25 I81 Verloop 825. [91 Davidson, J. F. and Harrison, D., F&idised Particles. Cambridge University Press 1963. [IO1 Clift R. and Grace I. R.. Chem. Engng Sci. 197227 2309. [III Upson P. C. and Pyle L., Proc. Znt. Symp. Fluidization Applications (Toulouse) p. 207, St& Chimie Industrielle, 1974.
1121Clift R., Grace J. R. and Weber M. E.. Znd. Engng Chem. Fundls 1974 13 45.
Cl31 Guedes de Carvalho J. R. F. Chem. Engng Sci. 198136 413. 1141Richardson J. F., Ruidkation (Edited by Davidson J. F. and Harrison D.). Academic Press, New York 1971. Ml Hovmand S. and Davidson J. F., Ruidizution (Edited by _ _ b~id~on 1. F. and Harrison D.). Academic Press, New York 1971. Guedes de Carvalho J. R. F., Ph.D. Thesis, Cambridge 1976. tr;; Guedes de Carvalho. J. R. F. and Harrison D.. Ruidised Combustion. Inst. Fuel Symp. Ser. No. I. 1975. [I81 Guedes de Cawalho J. R. F., King D. F. and Harrison D., Fluidisatian (Edited by Davidson 1. F. and Keains D. L.). Cambridge University Press, I9778. II91 Harrison-D., Davidson J. F. &xl de~Kock J. W., Trans. Inst. Chem. Emgm.London l%l 39 237. [201 Bagnold R. A., 77te Physics of Blown Sand and Lkseri Dunes. Metbuen, London 1954. [2I1 Clift R., FiIla M. and Massimilta L., Znr.L Multiphase How 1976 2 549. 1771
Yalin M. S.. Mechanics of Sediment Ttnn~port. Pergamon L-e’ Press, Oxford W72. r231 Inversen J. D.. Pollack J. B., Greeley R. and White B. R., Icarus 197629 381. [241 Zenz F. A.:Znd. Emgng Chem. Fundls 19643 65. WI Filla M., Davidson J. F., Bates I. F. and Eccles M. A., Chem. Enana Sci. 197631359. 1261 Persen L. p.1 Boundary Layer ‘ZReoty. Tapir, Forlag 1972. i27j Stewart P. S. B., Ph.D. Thesis, Cambridge 1%5. [281 Snbzwari M. P., Clift R. and Pyle D. L.. Z+ddh&ion (Edited by Davidson J. F. and Keairns D. L.). Cambridge University Press 197% 1291Varadi T. and Grace I. R.. Huidisation (Edited by Davidson J. F. and Keairns D. L.). Cambridge University Press 1978. 1301Chepil W. S., Soil Sci. 194560 397. 1311Chepil W. S., Soil Sci. Sot. Prvc. I959 23 422. [32] Zingg A. W.. Bull. 34. Univ. of Iowa Studies in Engng 1953. 1331Filla M., Ph.D. Thesis, Cambridge 1972.
BASED
ON PARTICLE
PICK-UP
FROM
THE
BEDS
WAKE
J. R. F. GUEDES DE CARVALHO Departamento de Engenharia Quimica. Faculdade de Eugenharia, Rua dos Bragas. Porto, Portugal (Received 1 August W30: accepted 22 lunlrary 1981) Abstract-A theory for slug stability in fluidised beds of fine particles is developed based on the idea of gas circulation inside bubbles. Data on the intensity of circulation inside gas slugs in liquids are reported and an analogy is drawn with slugs in fluidised beds; this information is coupled with data on gas pick-up velocity for flow over granular beds, to yield a simple criterion for slug stability. A test of this criterion is provided by comparison with data on the stabiity of freely slugging beds. An X-ray study of slugs in fluidised beds at various pressures is also described that gives further support IO the predictions
of the theory.
INTRODUCTION
It is well known that while most liquid-solid
systems fluidise homogeneously, the gas fluidisation of solids is usually aggregative with large gas bubbles passing through the bed. This behaviour is generally attributed to the low viscosity of gases and the large density difference between gas and solids. This idea gained some experimental support from the fluidisation of light solids with gases under pressure, which proved to be of the particulate type[l], and from the fluidisation of heavy solids with water, which proved to be aggregative[2]. If to the fluidising water was added some glycerol. thus increasing the viscosity, the state of uniform fiuidisation could be reached again, showing the effect of viscosity. Various workers tried to explain the origin of bubbles in fluidised beds. Rice and Wilhelm[3] have shown that the lower surface of a ffuidised bed supported merely by the flowing gas is unstable. Their analysis shows that a perturbation on the lower surface of a bed tends to grow, and that this process is a function of the effective density and apparent viscosity of the dense phase. Others[4-61 have made similar analyses and the general conclusion is that any small perturbation about the state of uniform fluidisation is initially magnified, giving rise to bubbles. According to these authors, the difference between gas and liquid fluidised beds arises from a much lower initial rate of growth of perturbations in the latter. Molerus[7] distinguishes between stable systems (liquid-solid type) and unstable systems (gas-solid type). However his conclusions, like those of the other authors, are of a qualitative nature. More recently Verloop and Hertjes[S] developed a criterion for the transition from homogeneous to heterogeneous fluidisation based on a comparison between the rise velocity of a porosity fluctuation and the longitudinal propagation of an equilibrium disturbance. However, their predictions are not always correct; in particular the influence of gas pressure as described by Leung[l], is not predicted.
Harrison et al. [2] have looked at the different types of fluidised behaviour in another way. For any given system they imagined a bubble rising through the bed and then proceeded to consider the stability of this bubble (a bubble in a liquid-solid system being a “liquid bubble”). The development of this idea has led to the concept of maximum stable bubble size. Systems in which the diameter of the largest stable bubble is comparable to the particle size fiuidise particulately and beds in which the largest stable bubbles are of a much larger size than the particles fluidise aggregatively. This theory provides a distinction between homogeneous and heterogeneous fluidisation in general agreement with experiment [9], and at the same time predicts the existence of a maximum size of stable bubble in systems in which bubbles form. An alternative theory which might explain a limitation in the size which bubbles can reach, is the theory of bubble breakup based on Taylor instability of the bubble roofflO-121. The application of this theory to the stability of fluidised beds of fine powders under pressure is considered in detail elsewhere [13]. TAETAEORYOFAARRtSONetsLANDlTS EXTENSIONTO SLUG STABILITY
Harrison et uf.[2] postulate the existence of gas circulation inside bubbles rising in fluidised beds, by analogy with gas-liquid and liquid-liquid systems; the gas inside the bubble is dragged down by the particulate phase (Fig. 5) and flows up in the middle. The maximum upwards velocity along the axis of the bubble is estimated to be UC= Lr, = 0.7bJ(gD.)
where iJb is the bubble rise velocity, g the acceleration of gravity and 0, the equivalent diameter to the bubble; the velocity UC is measured relative to axis fixed with the bubble. A solid particle is then imagined to be transferred from the wake to the inside of the bubble and
I349 CES Vol. 36. No. 8-F
(1)
1350
J. R. F.
GUEDES DE CARVALHO
consideration is-given to its possible behaviour: (i) if the upwards velocity, U,, is greater than the terminal free settling velocity of the particle, U,, the bubble tends to get filled with particles from the wake and the bubble is not stable; (ii) if iJ, < U, the particle falls back on the wake and the bubble is stable. For a given fluid particle system at given pressure and temperature, V, is fixed and because U, increases with D,, a bubble size exists for which U, = U, and this is the maximum size for a stable bubble in the system. In beds with a high ratio of height over diameter, H/D, slugging is often observed and the above theory on bubble stability may be adapted to the determination of slug stability. Experiments described in this paper suggest that the maximum upwards velocity in the slug, V,, is related to its rise velocity, U,, by U, = 0.9l_J, = 0.315d(gD)
(2)
Fig. 1. Criterion for bubble and slug stability based on of Harrison
Gas pressure in bar is indicated on the figure next to each point
cm9 gas
where D is the bed diameter. By analogy with bubble stability it may be said that slugging is not possible if CJ, < 0.315~(gD).
(3)
If a Reynolds number under free settling conditions is defined for a particle with diameter d, Re, =(&d/p) where p is the fluid density and CLits viscosity, it may be shown to depend on Ga only[l4]. Re, = f(Ga)
(4)
where GO = p(p, - p)gd3/p*, p. being the particle density. For spherical particles eqn (4) is given approximately by the following expressions (14) (in the form Gu = f-‘(Re,)) Ga = lgRe, Ga=18Re Ga =; Re:
(Ga < 3.6)
t +27Re’ . .687 I
(4.1) (3.6
(Gu >Q I@).
From(4) the condition thus be written as
(4.2) (4.3)
for bubble (or slug) stability may
UC < Wlpd)f(Gu).
(5)
Substitution of U, from (1) and (2) yields, after some algebraic manipulation
for bubble stability, and
for slug stability. Figure 1 shows these two conditions and it may be used to determine the maximum stable bubble size for a
the theory
et al. [2].
0.95x103 48 0.95x 103 82 2.90x I@ 64 2.9OxlV 64 2.92x I@ 68 2.69x t# 74 290x103 211
0.05 0.05 0.10 0.10 0.05 0.10 0.10
N2 N2
2,
Stable slugging
Unstable slugging
A E
$
c%* is N2
0
-o-
given fluid-particle system, by determining the relevant value of Ga and reading the corresponding ordinate off the bubble stability line. On the other hand, for a given gas-solid system in a tube both Go and the left band side in (5.2) may be determined, and the corresponding point plotted in the graph; slugging is predicted to be unstable for points above the solid line and stable for points below that line. In the extreme cases of free fall under laminar flow (Ga < 3.6) or fully turbulent flow (Ga > lo’), inequalities (5.1) and (5.2) lead to very simple analytical expressions, upon substitution of J(Go) by (4.1) and (4.2) respectively. Comparison of theory with experiment requires a clear understanding of the meaning of the stability of the slug. If the coalescence of small bubbles leads to the formation of a slug that is unstable, the foregoing theory predicts that it will break up by filling in from the lower surface. Evidence for slug instability in Ruidised beds may be sought from two sources; X-ray “viewing” of the bed (to be discussed later in this paper) and observation of bed surface fluctuation. In freely slugging beds the ratio of slug length to bed diameter increases with gas flowrate and height above distributor[M]; if slugs are unstable it is expected that the amplitude of surface fluctuations is only a fraction of D, even if the gas flowrate is high and the ratio H/D is large. This was the reasoning behind the method used by Guedes de Carvalho[l6] in order to study the influence of gas pressure on the stability of slugs. Detailed reports of the experimental findings are given elsewhere [ 17, 181 and the data relating to the stability of slugs are represented in Fig. 1. It may be seen that while the correct behaviour (stability of slugging) is predicted for glass particles
The stability of slugs in fluidised beds of fine particles
1351
211 pm in diameter fluidised by air, for very fine particles the impossibility of slugging is predicted, in contradiction with experimental observation[l7]. An explanation for this “misbehaviour” of beds of fine particles (< 100 Cm), based on the relative difficulty of pick up of particles from the wake has been suggested[19]; this is discussed in detail in the next section.
THESTABILITYOF THE LOWER SURFACE OF sLucs
Harrison
et ob[2] looked at the forces exerted by circulating gas on particles placed near the “center” of a bubble, but no consideration was given to particle transport from the bubble surface to this central position. A possible mechanism is that particles are picked up and entrained into the bubble when the drag exerted by the circulating gas over the lower surface of the bubble exceeds a certain value [ 191. Bagnold[20] studied the forces on a particle at the surface of a bed of sand across which air blows horizontally, as in a desert; the strength of the wind being measured by the drag velocity u* = g(~/p) where Q is the magnitude of the shear stress exerted by the wind on the surface. Bagnold found that there is a “threshold” drag velocity, (u,),, above which the horizontal drag on the particle is enough to overcome the opposed moment due to gravity, thereby causing the particle to ride over the neighbour against which it rests. This analysis may be applied to the stability of a slug (or bubble) in a fluidised bed, the surface of the desert in Bagnold’s analysis being equivalent to the lower surface of the slug and the wind over this surface being due to the vortex within the bubble. It should be stressed that the situation for which Bagnold’s analysis was developed does not match exactly the conditions over the lower surface of slugs. (i) Bagnold considered unidirectional gas movement over a sand bed and that is not the case inside the slug; however drag velocity is the important parameter and this can be calculated approximately for the flow conditions inside the slug. (ii) Gas percolation through the slug wake can be expected to help lift the particles; this may be true to some extent, but the force on a particle decreases markedly as it is lifted up from the bed on which it rests. Clift et ol.[21] applied Bagnold’s analysis to the analogous problem of pick-up of particles by a gas jet issuing into a fluidised bed and found good agreement between theory and experiment.
O.O11o
I 1OD
d@ml ’ lo
Fig. 2. Dependence of threshold drag velocity on particle size for sand in air (at room temperature), from eqn (6).
mechanism of particle pick-up, Bagnold [20] suggested expression for the threshold velocity (~1, = AX&$. - p)gdlp)
(6)
“constant”. Yalin[22] used where A is a dimensionless dimensional analysis to show that A depends only on (Re,), = p(u,),d/p. The relation between A and (Re*), has to be determined experimentally. Figure 3 shows most of the experimental data available, as given by Tnversen et af.[23] in a study concerned with aeolian processes on Mars. In that study a wide range of particle sizes and densities was covered and it may be seen that for (Re,), > 5 the value of A is constant al about 0.12. For the very fine particles (i.e. low (Re,),), interparticle forces may be important and it is possible that A is not a function of (Re*), alone[23]. Also shown in Fig. 3 are the data of Zenz[24] on saltation velocity, even though it refers to the slightly different situation in which single particles are picked up from a smooth horizontal surface. The values of (u*), for this case were computed from Blasius equation for turbulent flow in smooth pipes. The plot in Fig. 3 has the disadvantage of in&ding (u*), in both dimensionless groups and a more convenient plot of A vs Go is shown in Fig. 4 where only values corresponding to the solid line in Fig. 3 are reproduced. Values of (u,), for a given gas-solid system may be easily found from this plot.
Analysis of the “threshold velocity” The relation between threshold velocity
and particle diameter is illustrated for sand particles in Fig. 2. For large particles the drag velocity decreases with the square root of particle size until a diameter is reached (80 pm at 1 atm) below which the particles are totally immersed in the laminar sub-layer and increased threshold velocities are necessary to pick them up. This feature of very small particles could explain the failure of the theory of Harrison et a/.[21 to predict the behaviour of beds of fine powders. Based on a
an
a2
0.t
Fig. 3. Variation
of A with threshold
drag Reynolds
number.
1352
I
J. R. F. GUEDES DECARVALHO
. . . . ....4
0.1
1
.
.
. * . . ..
I
10
.
* . . . . .. 1
xl0
.
.
I.
i,doo &
plified analysis of the shear stress over the lower surface of the slug. The flow near the rim (Fig. 5) may be represented (as an approximation) by plane flow towards a stagnation point and this leads to a simple analytical solution for the drag velocity; the two dimensional simplification seems reasonable as the region where the drag velocity is highest is far from the axis of the slug. For plane flow towards a stagnation point the stream function is tj = kxy and the velocity parallel to the plane, ~O,L!OO u, varies with distance from the stagnation point, x, as
Fig. 4. Dependence of A on Galileo number.
u=kr
The plots in Figs. 3 and 4 are not applicable for liquid-solid systems as it has been reported[20] that for such systems the relation between A and (Rc,), is significantly different. At high (Re,),, the value of A for liquids is approximately double that for gases.
where k is a constant. The solution for the drag velocity over the surface is[261
Flow over the lower surface of a slug The analysis of slug stability based on particle pick-up from the wake requires the determination of the drag velocity over the bottom of the slug. If (I)*),,,*~ represents the maximum value of the drag velocity over the lower surface of the slug, particle pick-up and therefore slug instability should occur for (v*), <(l)+Jmax. For beds of fine particles the gas “throughflow” in the slug is negligible and an estimate of (u*),,,~~ may be obtained from data on circulation inside slugs in liquids. The two situations are indeed analogous. The analysis of flow around a slug[lS] shows that the particle velocity on the slug surface, I+,, is equal to the liquid velocity in the corresponding gas-liquid interface. Now the tangential component of the gas velocity v, (Fig. 5) has to be equal to u, otherwise a non zero pressure gradient apI& would show over the slug surface. The gas velocity perpendicular to the surface, u,,, is of order Umf and therefore negligible for systems of fine particles. As a result uplrtiFlc= vsnS over the slug surface and this is a “no slip” condition, similar to that observed on the gas-liquid interface. Filla et a!.[251 used ammonium chloride to help visualize the circulatory streamlines inside air slugs in water; a plot of the streamlines for a slug with length equal to twice the tube diameter was also obtained, based on a numerical solution of the equation of inviscid flow inside the slug. The shape of the streamlines suggests a sim-
u* = d(rdp)=
l.llu(Rex)-“4
(71
63)
where Re, = pux/p. The numerical solution for flow inside a slug does not give absolute values of velocity, but relative velocities between any two points may be calculated from it. In Fig. 6 the horizontal velocity close to the lower surface of the slug is given, relative to the maximum upwards centre line velocity, UC, as a function of distance from the rim, X. The velocity increases initially almost linearly with distance, reaches a maximum, and then decreases again as the axis is approached. The analogy with flow towards a stagnation point clearly ceases to apply at some distance from the rim. Figure 6 shows that the approximation is reasonable until x/D = 0.1, with ul U, = 7.4(x/D) at that point. If this is taken to be the point where the drag velocity is a maximum, substitution in eqn (8) gives (v*),.,
= 0.821 U,(l3.5,u/pDU,)“’
(9)
and all that is needed is a good estimate of UC. Filla et nl.[25] described a technique in which a slug is kept stationary inside an inverted rotameter tube by means of a downtlow of water, and a hot wire anemometer is used to measure centre line velocities inside the slug. In the present study, this technique was used to
X
Fig. 5. Idealised flow pattern near rim of slug.
Fii. 6. Dependence of horizontal velocity component on distance from rim for ideal flow inside slug &ID = 21.
The stability of slugs in fluidised beds of fine particles
is proportional to slug rise velocity the above value UC = 0.22 m/s suggests U, = 0.9U.. If this relation is assumed to hold for tube diameters other than that used in the experiment described, then
measure velocities inside slugs of various lengths, L,, held in a tube with average diameter D = 0.05 m. The experimental velocities are shown in Fig. 7 as a function of distance from the slug nose, z and the solid line corresponds to the theoretical values (i.e. numerical solution) with UC = 0.22 m/s. It may be seen that near the nose of the slug, theory and experiment are in good agreement; but as the bottom of the slug is approached a marked increase in gas velocity is observed. This is probably due to oscillation of liquid in the wake. In fluidised beds the dense phase is very “viscous” and such oscillations are not observed [27]. The velocity of rise of a slug in a tube of dia. D is U, = O.Xg(gD) and if the upwards circulation velocity
0.75
I
UC = 0.315d(gD) and substitution
0
= 0~~1)k18g318~lr4p-~~~~
(11)
Criterion for slug stabifity Equation (11) suggests that the maximum drag velocity over the bottom of a slug increases with tube diameter, although weakly, and since the threshold velocity for a given gas-solid system is fixed, there is a tube diameter for which (tl*),,, = (u,),, and slugs are not stable in beds with diameters larger than that. From (6) and (11) slugs are not stable if
a.rnn with
(10)
in (9) gives finally,
(u*),.,
c Cs-O.(6m 0 a.10 m o.vJm I
--Thy
1353
lJr=022mh
(Pa-;‘gdl”’
A
< 0.66 (F)II:
(12)
[
and this may be re-written
Fig. 7.
The possibility of slug instability increases with creasing pressure, whereas it is almost independent temperature as for most gases the product (pp) is sensitive to temperature changes, and is proportional pressure.
Gas circulation velocity along axis of “stationary” slug.
[
3
.
,,.I.’
s
.
10
.
. ..p
’
..,,*I
’
102
’
Id
’
.‘._
Ga
Fii. 8. Criterion for slug stability from theory in this paper Gas pressure in bar is indicated on the figure next to each point Stable Ref.
slugging
0.95 x 103 0.95x 1oJ 2.90x IO’
48 82 64
0.05 0.05 0.10
N, N, Nt
Ml ;;;;
n T
2.90x 2.92x 2.69x
64 68 74
0.10 0.05 0.10
co2 N, co2
tt:;
:
WI
0
LCP LO’ 10’
as
Unstable slugging
$ 0
inof into
J. R. F. GUEDES DECARVALHO
1354
Figure 8. shows a plot which may be used to determine the possibility of slug instability for a given gas-solid system in a tube. The full line corresponds to a plot of (A/0.66)’ vs Ga and as in Fig. 1, points above this line correspond to systems for which slugs are expected to be unstable, whereas for points below the line slugging should be stable: for points close to the line some degree of uncertainty is to be expected because of the approximations involved. For Ga > 3OMl the value of A is constant (A = 0.12) and slugs are stable if (13.1)
EXPERBIEWAL
EVIDENCE
Data on the behaviour of fluidised beds under pressure are scarce, but there is a widespread belief that fluidisation becomes “smoother” at higher pressures; “smoothness” being associated with the presence of smaller bubbles and therefore with smaller pressure fluctuations near the distributor. Bubbling bed behaviour Guedes de Carvalho and Harrison[l7] showed that fluidisation is not necessarily smoother at higher pressures, by Ruidising different powders in a narrow and long tube and observing their bubbling characteristics; al1 the powders were closely sized to avoid difficulty in the interpretation of the results. Synclyst particles (d = 48 pm) fluidised by nitrogen in a 0.05 m diameter tube could be made to slug at pressures up to 6 bar, but the bed level fluctuations decreased markedly in amplitude for higher pressures; this was taken as an indication that slugs were not stable at the higher pressures. The corresponding points are shown in Fig. 8 and they seem to agree with the criterion for slug stability presented above. The same authors fluidised glass hallotini (68 wrn) with nitrogen in the same tube and observed no change in bubbling characteristics at pressures up to 22 bar; at any pressure the bed could be made to slug easily if enough gas was supplied. In Fig. 8 it may be seen that this is in agreement with the theory. Another series of experiments [ IS] showed that 64 pm glass ballotini fluidised by carbon dioxide in a 0.1 m diameter bed showed unstable slugging for pressures above 20 bar, as predicted by the criterion for slug break-up. Figure 8 summarises the data available to the author on fluidisation of fine powders at high pressure and the agreement with theory of slug break-up seems to be very good especially if it is considered that the theory is based on a simplified physical model and no adjustable parameters were used to match theory and experiment. However, it would be desirable to have experimental data for other powders and at even higher gas densities. Bubble visualization An obvious way to test the theory would be to look into bubbles or slugs and see if particles were entrained from the wake. Subzwari et a[.[281 and Varadi and
Grace [29] photographed bubbles in two dimensional fluidised beds and observed that the dominant mechanism of bubble break-up was by indentation from the roof. However, this cannot be considered as refutation of the present theory because for the pressure range covered by those authors the present theory would predict large bubbles to be stable. 0n the other hand two dimensional beds are not the best tool to test the present theory because the walls are likely to interfere significantly with circulation inside the bubbles. An alternative method was used by Guedes de Carvalho [16] who used the X-ray facilities at University College. London, to film gas slugs in a 0.05 m cylindrical bed filled with catalyst. The X-ray beam and the tine camera were fixed at 0.8m above the distributor while the bed was fluidised by nitrogen: the gas flowrate was just enough for occasional slugs to appear and so avoiding frequent coalescence in the section viewed. Exposure times were 1/125Os per frame and sequences of five consecutive frames are shown in Figs. 9(a) and (b). Runs 1 and 2 were taken at 15 frames/s and Runs 3,4 and 5 were taken at 20 frames/s. The pressures and the characteristics of the particles used in the X-ray studies are given in Table 1. Runs 1, 3 and 4 were made at pressures below those required for slug instability, as predicted from the present theory, and the lower surface
Run 1
Run Z(a)
Run 2(b)
Fig. 9(a). X-Ray pictures of rising slugs.
i355
The stability of slugs in fluidised beds of fine particles
boundaries and shatter on occasions); the pressure at which this occlirs is predicted with good approximation by the theory developed in this paper. CONCLUSIONS
theory of Harrison et a/.[21 on bubble stability is shown to disagree with experimental results for fluidised beds of fine powders. An extension of that theory is considered based on the phenomenon of pick-up of particles from a granular bed and on an approximate description of circulatory gas flow inside a slug. The resulting quantitative criterion of slug stability is shown to be in good agreement with available experimental data. The same criterion defines a range of experimental conditions for which data should be sought in order to check further the theory in this paper. The
Acknowledge~enrs-Part of this work was done while the author was at the Department of Chemical Engineering, University of Cambridge. Financial support from Calouste Gulbenkian Foungratefully acknowledged. NOTATION
Run 3
“constant” in eqn (6) particle diameter or average particle diameter bed diameter equivalent diameter of bubble acceleration of gravity Galileo number bed height constant, eqn (7) slug length PU, d/cl P(U& d/p
Run 5
Run 4
Fig. 9(b). X-Ray pictures of rising slugs.
PUlfi
coordinate along slug surface velocity along wall in flow towards a stagnation point; horizontal velocity over lower bubble surface bubble rise velocity maximum upwards velocity along bubble axis, relative to axis fixed with the bubble slug rise velocity free settling velocity of particles drag velocity over surface maximum drag velocity
of the slugs may be seen to be reasonably flat. In the sequence corresponding to Run 4 in particular, an indentation develops from the roof and plunges into the wake, but leaves it unaffected. However. in Runs 2 and 5, the pressure is slightly higher than required for the slugs to be unstable according to the present theory, and the lack of definition of the slug boundaries (the wake included) is particularly noticeable. Although these pictures do not elucidate the exact mechanism of slug instability, they show nevertheless that above a certain pressure slugs become unstable, (i.e. have ill defined
Table I. Experimental conditions in X-ray experiments Run
Pressure
(bar) 1
1
2
(a>
14
2
(b)
14
Ps
(kg/m3 1
d (Inn)
950
75
950
75
950
75
3
1
950
59
4
6
950
5s
5
12
950
59
J. R. F. GIJEDES DECARVALHO
1356
threshold
drag velocity perpendicular to slug surface particle velocity over slug surface gas velocity parallel to slug surface distance from plane of symmetry in flow towards a stagnation point; horizontal distance from rim, Fig. 5 distance from wall in flow towards a stagnation point; distance from lower slug surface distance from slug nose fluid viscosity fluid density particle density shear stress stream function gas velocity
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Leung L. S., Ph.D. Thesis, University of Cambridge l%l.
;;; Harrison D., Davidson J. F. and de Kock J. W., Tmns. Inst. Chem. Engrs London 1961 39 202. [31 Rice W. I. and WiIhelm R. H., A.Z.Ch.E.Z. 1958 4 423. [41 Jackson R., Trans. Inst. Chem. Engrs London 1963 41 13. Murray J. D., J. Fluid Me&. I%5 21 465. :; Pigford . ^_ R. L. and Baron T., Ind. Engng Chem. Fund/s 1965 4 61. [71 Molerus, 0.. Proc. intern. Symp. Fiuidization (Edited by Drinkenburg A. A. H.) p. 134, Netherland University Press, Amsterdam 1967. 1. and Heertjes P. M., Chem. Engng Sci. 1970 25 I81 Verloop 825. [91 Davidson, J. F. and Harrison, D., F&idised Particles. Cambridge University Press 1963. [IO1 Clift R. and Grace I. R.. Chem. Engng Sci. 197227 2309. [III Upson P. C. and Pyle L., Proc. Znt. Symp. Fluidization Applications (Toulouse) p. 207, St& Chimie Industrielle, 1974.
1121Clift R., Grace J. R. and Weber M. E.. Znd. Engng Chem. Fundls 1974 13 45.
Cl31 Guedes de Carvalho J. R. F. Chem. Engng Sci. 198136 413. 1141Richardson J. F., Ruidkation (Edited by Davidson J. F. and Harrison D.). Academic Press, New York 1971. Ml Hovmand S. and Davidson J. F., Ruidizution (Edited by _ _ b~id~on 1. F. and Harrison D.). Academic Press, New York 1971. Guedes de Carvalho J. R. F., Ph.D. Thesis, Cambridge 1976. tr;; Guedes de Carvalho. J. R. F. and Harrison D.. Ruidised Combustion. Inst. Fuel Symp. Ser. No. I. 1975. [I81 Guedes de Cawalho J. R. F., King D. F. and Harrison D., Fluidisatian (Edited by Davidson 1. F. and Keains D. L.). Cambridge University Press, I9778. II91 Harrison-D., Davidson J. F. &xl de~Kock J. W., Trans. Inst. Chem. Emgm.London l%l 39 237. [201 Bagnold R. A., 77te Physics of Blown Sand and Lkseri Dunes. Metbuen, London 1954. [2I1 Clift R., FiIla M. and Massimilta L., Znr.L Multiphase How 1976 2 549. 1771
Yalin M. S.. Mechanics of Sediment Ttnn~port. Pergamon L-e’ Press, Oxford W72. r231 Inversen J. D.. Pollack J. B., Greeley R. and White B. R., Icarus 197629 381. [241 Zenz F. A.:Znd. Emgng Chem. Fundls 19643 65. WI Filla M., Davidson J. F., Bates I. F. and Eccles M. A., Chem. Enana Sci. 197631359. 1261 Persen L. p.1 Boundary Layer ‘ZReoty. Tapir, Forlag 1972. i27j Stewart P. S. B., Ph.D. Thesis, Cambridge 1%5. [281 Snbzwari M. P., Clift R. and Pyle D. L.. Z+ddh&ion (Edited by Davidson J. F. and Keairns D. L.). Cambridge University Press 197% 1291Varadi T. and Grace I. R.. Huidisation (Edited by Davidson J. F. and Keairns D. L.). Cambridge University Press 1978. 1301Chepil W. S., Soil Sci. 194560 397. 1311Chepil W. S., Soil Sci. Sot. Prvc. I959 23 422. [32] Zingg A. W.. Bull. 34. Univ. of Iowa Studies in Engng 1953. 1331Filla M., Ph.D. Thesis, Cambridge 1972.