Stability of Bubbles in Fluidized Beds R. Clift, J. R. Grace,* and M. E. Weber Department of Chemical Engineering, McGill University, Montreal, Canada
a bubble in a fluidiz’ed bed leads to predictions of initial growth rates and most sensitive wavelengths for disturbances of t h e interface. It is shown that the degree of instability is primarily a function of t h e effective kinematic viscosity of the dense phase, while t h e interstitial fluid velocity has a comparatively minor influence. T h e results are consistent with available evidence on the incidence of bubble splitting in fluidized systems. A linearized stability analysis of the upper surface of
Introduction Bubble splitting is one important factor determining the bubble size distribution and hence the effectiveness of fluid-solid contacting in gas-fluidized beds. In a recent note (Clift and Grace, 1973), it has been shown that the first stage of bubble splitting is development of an indentation in the bubble roof. Such an indentation moves around the periphery of the bubble, simultaneously growing to form a “stalactite” or “curtain” of particles. Splitting occurs if the lower edge of the curtain reaches the base of the bubble before the top passes the equator. It was suggested that this phenomehon results from instability of the type discussed by Taylor (1950), where a heavy fluid overlies a lighter one. Henriksen (1972) has presented experimental evidence which leads to the same qualitative conclusion for three-phase fluidized beds and conventional two-phase systems. In the present paper, linearized stability analysis is applied to predict the conditions under which a bubble may split by growth of a dividing curtain. The treatment differs from earlier analyses (Porter, 1963; Rice and Wilhelm, 1958) both in the treatment of the particulate phase and in the aim of the analysis. Rice and Wilhelm discussed the influence of Taylor stability on bubble splitting, but the emphasis of both earlier papers is on bubble formation. In each case, it was assumed that the distributor is blanketed by a layer of the fluidizing fluid and that bubbles are generated by instability a t the boundary of the overlying dense phase. There is no direct evidence to suggest that bubbles are actually formed by this mechanism. Particularly with a distributor of the sieve-plate or tuyere type, fluid enters the bed as jets which penetrate into the dense phase and form bubbles (Fakhimi and Harrison, 1970; Zenz, 1968). Thus it appears that Taylor stability analysis should more properly be directed at the bubble splitting problem.
their models, Rice and Wilhelm (1958) used a Newtonian viscosity well in excess of that for the fluidizing fluid alone. In the present case, the particles and fluid comprising the dense phase were considered separately by treating the dense phase as two superimposed, continuous fluids (see Anderson and Jackson, 1967). The interstitial fluid percolates upward with absolute velocity Wi in the dense phase and occupies a volume fraction t at any point; the remainder of the dense phase is occupied by the “particulate fluid.” The scale of the phenomena under consideration is assumed to be large by comparison with particle dimensions, so that the velocity in each dense phase fluid and the volume fraction t can be treated as continuous functions. Formally, voidage variations were permitted in the manner used by Jackson (1963); i . e . the density of the dense phase was assumed constant, but its permeability (which depends much more strongly on voidage) was permitted to vary. However, it turns out (see Appendix A) that this implicit treatment of voidage effects cannot be applied to the stability analysis since the assumption of negligible density variation eliminates variation in the fluid-particle interaction coefficient. The separate treatment of particles and fluid in the dense phase requires boundary conditions which differ from those in the earlier studies and these are derived in Appendix B. To solve the governing equations it is assumed that an infinitesimal disturbance on the interface grows as exp(nt). The analysis leads to three distinct equations for the evaluation of n. If the viscosity and density of the fluidizing fluid are very much smaller than the corresponding quantities for the particulate phase ( e . g . , for gas fluidized beds) the analysis gives
R4
2R2
-
4R
+
(1
-
g/k%,2) = 0
(1)
where k is the wave number of the disturbance
R =
Growth of a Disturbance The basic physical situation to be examined is shown in Figure 1. A dense phase consisting of solid particles and interstitial fluidizing fluid lies above particle-free bubble phase. In the unperturbed state, the interface is assumed to be horizontal and stationary a t x = 0. For simplicity the treatment is two-dimensional throughout. Appendix A summarizes the development of governing equations of motion and continuity in forms which are linear in parameters describing a disturbance to the unperturbed state. In the earlier analyses of this problem the dense phase was treated as a single incompressible fluid. Porter (1963) assumed that the dense phase had the same shear viscosit y as the fluidizing fluid while, in the more realistic of
+
,/*
and
(3) is the apparent kinematic viscosity of the particulate phase. Note that the only parameter determining the stability of the interface for this simplified case is the apparent kinematic viscosity of the particulate phase, vp. In fact, eq 1 is identical with that obtained by Bellman and Pennington (1954) for real fluids with negligible interfacial tension and with the upper fluid much more dense . and viscous than the lower. The real positive root for R can be obtained by standard numerical techniques. If the density and viscosity of the fluidizing fluid are retained, two possible cases must be considered. The first Ind. Eng. Chem., Fundam., Vol. 13, No. 1, 1974
45
A. cm
!nterstitial fluid
DENSE PHASE
{Parriculate
fluid
1.0
0.5
80 fluid velocity BUBBLE PHASE
Figure 1. Schematic diagram of unperturbed system Table I. Representative Fluidized Systems”
System Particle density, pP, g/cmE Dense phase kinematic Viscosity, vP, cm2/sec Fluid density, Pf, g/cm3 Fluid viscosity, Pf,p
Interstitial velocity, Wi, cm/sec
1
2.5
2
11.4
3 2.5
4.0-10.0
0.5-2.0
0.1-1.0
1.26 X
1.0
1.0
1.84 X 10-4
0,0105
0.0105
1.0-100.0
2.0-20.0
0.005-0.5 k.cml
is taken as 0.4 and acceleration due to a Voidage (e) gravity (g)as 981 cm/sec2 throughout.
Figure 2. Growth rates predicted by simplified equation, eq 1, neglecting density and viscosity of interstitial fluid
applies when the momentum of the interstitial fluid is negligible compared with that of the particles, and leads to eq C11. When the interstitial fluid momentum is included, eq C14 results. These two equations will be termed the “intermediate” and “complete” equations, respectively, while eq 1 will be called the simplified equation. The three equations have been solved numerically for three fluidized systems whose properties are given in Table I. System 1 represents sand particles or glass beads of various sizes fluidized by air. The range of effective dense-phase viscosity is inferred from the shape of bubbles in such systems (Grace, 1970). Systems 2 and 3 correspond to water-fluidized beds of lead shot and sand, respectively. The former is known to behave in an aggregative manner; system 3 shows particulate behavior, but is included to indicate whether the stability analysis can account, for the absence of bubbles in such systems. The range of effective viscosities for the two water-fluidized systems was estimated from the semiempirical correlation of Hetzler and Williams (1969). Predictions from the simplified equation are presented in Figure 2 for effective kinematic viscosities covering the complete range shown for the three systems in Table I. The curves are drawn only to wavelengths approaching typical particle dimensions in view of our assumption that the dense phase behaves as a continuum. Increases in the dense phase kinematic viscosity are seen to result in a reduction in the growth factor and in the “most sensitive” or “most dangerous” wave number, at which n is a maximum. For system 1 where pp >> pf and pp >> p f , the predictions of the complete and intermediate equations agree very closely with those of the simplified equation. Even for Wi = 100 cm/sec, the predicted growth factors agree within 0.5%. Thus for gas-fluidized systems, the interstitial gas velocity has virtually no effect on the stability of a bubble roof whereas the effective kinematic viscosity of the dense phase plays the determining role.
For liquid-fluidized systems (systems 2 and 3) the differences between predictions of the simplified equation and the more complete equations are more significant. Both the complete and intermediate equations still predict positive growth factors, but their values are smaller than the results shown in Figure 2 for the simplified equation. For both systems 2 and 3, the value of c, which specifies the tangential boundary condition (see Appendix B), generally causes less than 2% variation in n. For system 2, where there is a large density difference between the lead particles and the fluidizing water, the intermediate and complete equations generally agree within 4% so that the neglect of interstitial fluid momentum has little effect on the predicted growth factor. Predicted growth factors are plotted in Figure 3 for system 2 based on the complete equation for two values of Wi and on the simplified equation. The curves and the most sensitive wave number depend strongly on v p but only weakly on the interstitial liquid velocity. For system 3 the difference between the complete and intermediate equations is somewhat more significant as shown in Figure 4. In this case there is virtually no effect of the magnitude of the interstitial liquid velocity: increasing Wi from 0.005 to 0.5 cm/sec decreases n by less than 1%. In the vicinity of the most sensitive wave number, the growth factor is now 40-45% lower than predicted by the simplified equation, but it is still larger than for systems 1and 2 due to the lower kinematic viscosity. These results show that the simplified equation (eq 1) may be used to predict most sensitive wavelengths and initial growth rates for gas fluidized beds. For liquidfluidized beds, the simplified equation still gives a good indication of the most sensitive wavelength but one of the more complete equations (eq C11 or C14) should be used to estimate growth factors for the early linear growth stage. Since the intermediate equation is simpler and the interstitial fluid momentum has little influence, eq C11 is recommended. In each case, the effective kinematic viscosity of the dense phase is the most important factor in
46
Ind. Eng. Chem., Fundam., Vol. 13, NO. 1, 1974
A ,cm 50
5
10
20
A, c r r
2
0.5
1.0
I 1.o
I
10
5
1.0
2
!OO, ,
0.5
1 20
k, crfi'
Figure 3. Growth rates for system 2 (lead-water) predicted by complete equation (eq C14): A, u = 0.5 cm2/sec; B, u p = 2.0 cmz/ sec; 1, simplified equation (eq 1);2, complete equation, W, = 2.0 cm/sec; 3, complete equation, Wi = 20:O cm/sec
determining initial growth rates and most sensitive wavelengths. Application to Bubble Splitting The present analysis predicts that the roof of a bubble in a fluidized bed is unstable to the growth of disturbance of all wavelengths. However, as noted in the Introduction, disturbances initiated on the roof of a bubble are swept around the periphery so that in practice a bubble does not split unless the disturbance has grown sufficiently before the tip of the growing spike reaches the side of the bubble. An estimate of the likelihood of splitting may be obtained by comparing the time required for a disturbance to grow by a given factor with the time available for growth. An order-of-magnitude estimate of the required growth time is provided by t e = n-1; i.e., t e is the time required for a small-amplitude disturbance to grow by a factor e. For gas-fluidized beds n may be estimated from eq 1 and 2. Although this procedure is no longer accurate for disturbances which have grown beyond the scale described by the linearized analysis, the estimate t e = is retained in the absence of any alternative. An order-of-magnitude estimate of the time available for growth, t a , may be obtained from the dense phase tangential velocity at the interface in the absence of any disturbance. This velocity may be estimated from potential +$ and 2J3 in two and three flow theory taking &/*as dimensions, respectively (Davidson and Harrison, 1963). Thus ~
UO
sin 0
=
(4)
for both two- and three-dimensional bubbles. If the disturbance originates at 81, the time available for growth is then approximately
t,
=
RoS'J2dO/U, 91
i.e.
ta =
E
In [tan
(5)
Thus the time available for growth becomes large if the disturbance originates very close to the nose. Observations
"
'0
1.0
30
k, crii'
Figure 4. Growth rates for system 3 (sand-water) predicted by intermediate equation which neglects the momentum of the interstitial fluid (eq Cll) and by complete equation (eq C14):A, y P = 0.1 cm2/sec; B, up = 1.0 cm2/sec; 1, simplified equation (eq 1);2, equation neglecting interstitial momentum (eq C11);3, complete equation (eq C14);Wi = 0.05cm/sec in each case
of splitting bubbles suggest that disturbances usually develop in a regular pattern to either side of the nose. We may therefore consider two distinct cases. Case A. The bubble nose is a node when the disturbance originates; Le.
81 = XJ4Ro = T / ~ ~ R Q (6) Case B. A node is located X/4 from the bubble nose so that the nose is an antinode in the initial disturbance; i.e.
h / 2 R ~= TJkRo (7) Substitution of eq 6 or 7 into eq 5 yields the maximum time available for growth, t a m . In case A the disturbance is assumed to originate closer to the bubble nose than in case B, thus yielding longer times. In each case. the available growth time decreases as the wavelength increases. The likelihood of splitting may now be assessed by comparing the values of t e and t a m , as shown in Figure 5. A bubble is liable to be split by a disturbance for which t a m > t e . The chain-dotted curves show values of t a m for 2and 5-cm radius bubbles. Only disturbances with wavelength less than the arc length from the nose to the equator, X I X m a x = nRo/2, are considered since a disturbance with wavelength greater than X m a x represents a gross deformation of the bubble rather than a perturbation on the interface. The solid lines in Figure 5 show the time required for growth, t e , evaluated from eq 1 for apparent kinematic viscosities of 10 and 4 cm2/sec. For short wavelength disturbances, t e exceeds t a m . Thus, even though the stability analysis predicts that all disturbances grow, disturbances of small wavelength grow so slowly that they do not achieve an amplitude large enough to cause splitting before they are swept around to the bubble equator. Longer wavelength disturbances demonstrate the importance of apparent kinematic viscosity. For example, consider case B for a bubble with Ro = 2 cm. In a bed with v P = 4 cm2/sec, wavelengths in the range 0.35 to 3.1 cm show t a m > t e S O that a disturbance in this range may cause splitting. However, the same bubble in a bed with
61
=
Ind. Eng. Chern., Fundarn., Vol. 13, No. 1, 1974
47
c
, &OX
I
1-
00102
max
10
10
x
40
.m
Figure 5. Comparison of maximum time available for growth, tam, with time, t,, for a distrubance to grow by a factor e (system 1)
A, = 10 cm2/sec always shows t a m < t e and therefore should not split whatever the wavelength of the disturbance. By contrast, a bubble of radius 5 cm in either bed is liable to be split by a broad range of disturbance wavelengths. It may also be noted that the most sensitive wavelength (minimum t e ) does not correspond to the wavelength most likely to cause splitting. The most sensitive wavelength may exceed Amax, as for case A with a 2-cm radius bubble in a bed having u p = 10 cm2/sec. When the most sensitive wavelength is within the range of possible disturbances, the ratio tam/te is a maximum for a wavelength less than the most sensitive.
Discussion More quantitative comparisons of t a m and t e cannot be made without additional information on the initial amplitude of the disturbance and the angle at which the disturbance originates. It must be recognized that the quantitative estimates of this model give only an order of magnitude for the size of bubble which is expected to be stable and free from splitting. In order to make the problem tractable a number of simplifications were introduced ( e . g . , the unperturbed interface was assumed horizontal and two-dimensional) and we have considered only the linear stage of growth. Nevertheless, the qualitative model does describe the correct mechanism of splitting. We now show that the analysis provides a basis for interpretation bf what few data are available on bubble break-up. An important conclusion of the analysis is that the effective kinematic viscosity of the dense phase is the dominant factor determining the initial growth of instabilities and the most dangerous wave number. Thus prediction of the effect of system properties on bubble stability depends on prediction of the effect on u p . The important parameters are as follows. (a) Mean Particle Diameter. Experimental measurements (see Schugerl, 1971) indicate that for the particles with diameter less than about 100 p, u p increases as the mean particle diameter (dB)increases; for larger particles, there appears to be no systematic effect of d, on u p . This dependence is consistent with the correlation of Hetzler and Williams (1969) which predicts that Y, is proportional to Wild,. The stability analysis therefore suggests that the initial growth rate of a disturbance should decrease and the most dangerous wavelength should increase with increasing d, up to a size of about 100 p . Thus bubbles should split most readily in small particle systems, and this is consistent with observed “maximum stable bubble 48
diameters” (Matsen, 1973) which increase in magnitude with increasing particle diameter. (b) Particle Density. There is some evidence (Schugerl, 1971) that Y, increases as particle density increases. This would result in bubbles being stable to larger sizes for denser particles, which is consistent with the commonly held belief that fluidization becomes more aggregative as the ratio p p / p f is increased. (c) Particle Size Distribution. Schugerl (1971) indicates that addition of fines to a bed of particles of narrow size distribution causes the bed viscosity to increase and pass through a maximum. However, Matheson, et al. (1949), present exactly contradictory evidence. Interpretation of the effect of size distribution in terms of the present model is therefore not clear. There is a general belief that addition of fines to fluidized beds leads to “smoother” fluidization (presumably smaller bubbles), but as Geldart (1972) has pointed out, this may simply result from the lowering of the mean particle diameter (see (a) above) rather than from the breadth of the particle size distribution. (d) Absolute Pressure in Gas-Fluidized Systems. Operation of gas-fluidized beds a t elevated pressures is known to give smaller bubbles and a more nearly particulate behavior. At first sight this is inconsistent with our analysis which predicts virtually no effect of p f provided that it remains much smaller than p,. However, it is unclear whether the observation of smaller bubbles refers to equal mass flow rates of gas or to equal superficial gas velocities. Thus, smaller bubbles in high-pressure systems may well result from smaller bubble flow rates. The analysis given in this paper is therefore consistent with the meager experimental evidence available on bubble splitting. More data on the relative frequency of splitting in different systems and on the influence of various factors on dense phase viscosities are required to test the theory more fully. The present results may be related to the problem of whether a given fluidized system shows aggregative or particulate behavior. Somewhat larger growth factors are predicted for disturbances in systems which are known to show particulate behavior (e.g., system 3 in Table I); the difference results from lower values of vP for such systems. However, the difference appears to be too small to explain the distinction between particulate and aggregative behavior. Evidently the essential difference between the two types of behavior is the degree of instability of the particulate phase to voidage disturbances, since this determines whether bubbles form in the first place (Jackson, 1971). It is noteworthy, however, that when a fluidized system expands in a particulate manner, the increase in voidage causes a marked decrease in the effective kinematic viscosity (Hetzler and Williams, 1969). Therefore, it is expected that as a bed undergoes particulate expansion, bubbles would become less and less stable. Thus, a particulate bed secures itself against stable bubbles. In this sense, then, the present work complements the stability analyses aimed at the problem of bubble initiation in fluidized systems. One further conclusion emerges from the mechanism of bubble splitting discussed in this paper. This is that instead of a discrete “maximum stable bubble size” below which all bubbles are stable, we expect bubble splitting to occur over a relatively broad and continuous range of bubble sizes. Whether or not a particular bubble splits will depend not only on its size but also on the occurrence, angular position, wavelength, and amplitude of disturbances of the bubble interface. It seems likely that measured maximum stable bubble diameters correspond to mean
Ind. Eng. Chem., Fundarn., Vol. 13,No. 1, 1974
diameters for systems in which a dynamic equilibrium has been achieved between coalescence and splitting. Conclusions A bubble in a fluidized bed splits when a developing curtain of particles reaches the floor of the bubble before the top of the curtain is swept around to the equator. A linearized analysis of the stability of the roof of a bubble shows that the interface is unstable to disturbances of all wavelengths. However, disturbances of very small wavelength grow relatively slowly and are swept to the bubble equator before they have grown sufficiently to cause the bubble to split. For large bubbles there is a range of wavelengths which tends to cause bubble splitting. The initial rate of growth of a disturbance, and hence bubble stability, depend strongly on the effective dense phase kinematic viscosity. The properties and the interstitial velocity of the fluidizing fluid have negligible influence for gas-fluidized beds and a small influence on stability for liquid-fluidized systems. The results of the analysis provide a consistent framework for interpreting available data on bubble splitting and for explaining the influence of such factors as bubble size, particle diameter, and particle density. Acknowledgment Financial assistance from the National Research Council of Canada is gratefully acknowledged. The assistance of Miss C. Cordon with the computational work is also greatly appreciated. Appendix A. Linearized Equations Since the analysis used in this work is qualitatively similar to those presented by Rice and Wilhelm (1958) and Porter (1963), only the essential features of the development are given. The velocity of each fluid is considered as the sum of a steady value and a perturbation velocity. For the particulate fluid, the steady velocity is zero. For the interstitial and bubble fluids, the instantaneous velocities are
Ui
= Wii
+
+ (v.v)v
= -gi
+
pp
Pp(1 -
v2v
€1
= pp(l
ax
ay
i n t e r s t i t i a l fluid
b u b b l e fluid
Differentiating the x component of eq A3 with respect to x, the y component with respect to y, adding, and using eq A7 and A8 to eliminate terms, we obtain
Since the disturbance is limited to the region of the interface, perturbation parameters vanish at large positive or negative x. In particular. t 0 as x m , so that eq A10 implies z = 0. Thus the assumption of negligible density variation in the dense phase eliminates any variation of the fluid-particle interaction coefficient. After omission of terms of second order in the perturbation parameters, the equations of motion become particulate fluid
- -
i n t e r s t i t i a l fluid
b u b b l e fluid
+ where Pi = pi-+ b p ( l - €1 -k pft]gX and P b = pb - P + ppgx, with P the pressure at the interface in the unperturbed state. Equations A7-A9 and All-A16 are the governing linear equations. Only eight of these equations form an independent set since they have already been used to eliminate z .
P
while for the interstitial fluid
(-44 1
In effect, it is assumed that pressure gradients in the dense phase are transmitted by the fluidizing fluid while shear stresses arise predominantly from particle-particle interactions. The latter are accounted for, to a first approximation, by ascribing an effective Newtonian shear viscosity to the particulate fluid. The fluid-particle interaction coefficient, 2, is also written as the sum of steady and perturbation terms
z = 2 + z
-
(A6) t)g Continuity equations for the three fluids, assuming both phases to be incompressible, are particulate fluid
2Wi
(AI)
ui
For the bubble phase, the equation of motion is simply the Navier-Stokes equation. Equations of motion for the dense phase fluids have been the subject of considerable controversy (Van Deemter, 1967). The equations adopted here have the virtues of simplicity and consistency. For the particulate fluid we write at
In the unperturbed state, eq A4 becomes
(A5)
Appendix B. Boundary Conditions The free boundary condition a t the interface, x = y(u,t), is
Equality of normal and shear stresses a t the interface requires Ind. Eng. Chem., Fundam., Vol. 13, No. 1, 1974
49
These equations are treated exactly as in the derivation of eq C3 from eq A15 and A16 to yield u, = en( sin ky(M,e+ M8-m~)
From continuity in the x direction we write ubx
= u,(l
-
e)
+
+
034)
fuix
For the tangential velocity condition, it is probably most reliable to relate the fluid velocity in the dense phase to the velocity gradient in the bubble phase (Beavers and Jcseph, 1967; Beavers, et al., 1970; Saffman, 1971). However, since the boundary condition on the y components of velocity turns out to have very little influence, a simpler formulation giving a linear relationship between uby, u y , and u i y is preferred; i. e.
1
u y = -xent
uiy
1
3e
ubx = Ne"'g(x) sin k y
(C2)
cos k y [ M , k ( l
-E@'
Substituting eq C3, C7, and C9 into the boundary conditions, eq B2 to B5 evaluated on the interface defined by eq C10, yields a set of four homogeneous linear equations in MI, M z , N1, and Nz. For nontrivial solutions to exist, the determinant of the coefficients in these equations must vanish, Le. 1 tnWJg 1 -1 -1
+ + cnWi/g) 2&'
b u b b l e fluid
m pp(m2
D
where k is the wave number of the disturbance and n is the growth factor. Substitution of eq C1 and C2 into eq A7 and A9 yields expressions for v y and u b y . For the bubble fluid, eq A15 and A16 are differentiated and subtracted to remove P b , and then ubx and uby are substituted to yield a fourth-order ordinary differential equation in g(x), Using the condition that ubx and uby are finite as x m, we obtain
-
+
N,le'x)
+ a1
cb,)
(1
+
cb2)
a2
Ind. Eng. Chem Fundam., Vol. 13,No. 1, 1974
(1
k 1 k') -2pfk2 -pdk2
E
D
(pp
E
F
+
P)
-2d = 0 (Cll)
-
pfXl
= (p,
F
=
- t ) g / n - pp(1 - c)n/k - 2pph - p d l - c ) g / n - 2ppm (a) p&/k
+
cWJ
-
2pfk
The second and fourth terms in the last row have been simplified using eq C4 and C8. Equation C11 enables n to be calculated as a function of k for given fluid and particle properties. For most gas-fluidized beds pr
