GREEKLETTERS 6,s = Kronecker delta 7 = 6Fz/[743(P, - P f k l Pf = fluid density PP = particle density 4 = fluctuation phase lag between particle velocity and fluid velocity W = T U 2L (frequency of fluid velocity fluctuations) SUBSCRIPTS
i
= zth component = vertical component = vertical plane
2
U
literature Cited Barker, 3. J., D. Eng. Sc. thesis, New York University, New
York, 1959. Barker, J. J., Treybal, R. E., A.I.Ch.E. J . 6, 289 (1960). Calderbank, P. H., Moo Young, M. B., Chem. Eng. Sci. 16, 39 ( 1 9 h- l-’/i . \
-
Harriot, P., Chem. Eng. Sci. 17, 149 (1962). Hegge-Zijnen, B. G. van der, Appl. Sci. Res., Sect. A7, 205 (1958). Hinze, J. O., “Turbulence,” p. 352, McGraw-Hill, New York, 1050
K&ske, .\. A , Pien, C. L., Ind. Eng. Chem. 36, 220 (1944). Kim, \l‘. J., Manning, F. S., A.I.Ch.E. J . 10, 747 (1964). Mack, D. E., Marriner, R. .\., Chem. Eng. Progr. 45, 545 (1949). Maisel, D. S., Sherwood, T. K., Chem. Eng. Progr. 46, 172 (1950). Marancozis. J.. Johnson. .A. I.. Can. J . Chem. Ene. 40. 231 (1962). Nagat< S., ’et a’l., M e m . Fac. E&. K2oto Univ. 22,“86 (i960). Oyama, Y., Endoh, K., Kagaku Kogaku 20, 576 (1956). Ranz, \Y.E., Marshall, \Y. R., Jr., Chem. Eng. Progr. 48, 141, 173 ,
I
/,ncn\
(lY>L/.
Rouse, H., Proceedings, Fifth International Congress for Applied Mechanics, p. 550, Ij’iley, New York, 1939. Schwartzberg, H . G., Ph.D. thesis, New York University, New York. 1965. Schwarizberg, H. G., Proposal to Federal \l’ater Pollution Control Administration, 1967. Schwartzberg, H. G., Treybal, R. E., IND.ENG. CHEM.FUNDAMENTALS 7; 1 (1968). Steinberger, R. L., Treybal, R. E., A.I.Ch.E. J . 6, 227 (1960). Tchen, C. M., Ph.D. thesis, Delft, 1947. ’
~
Comings, E. \Y.,Clapp, J. T., Taylor, J. F., Ind. Eng. Chem. 40, 1076 (1948). Cutter, L. . 0. Then, Ivith the positive sign before the square root, Equation 31 is seen to give a positive value for 7, showing that the velocity of propagation is in the upivard vertical direction. I t is not possible to say very much about the dependence of and 7 on k , in general without numerical computation, but the limiting forms of these quantities at large and small values of the Lvave number can be written.
and
Ivhere ~ e ’=
Comparing this with Equation 19 it is seen that terms in the virtual mass coefficient are smaller in 19a, so the results are more sensitive to the value of the virtual mass coefficient when the theory is based on Equation 19. For this reason we have chosen to develop Equation 19 rather than 19a, since a theory based on 19 provides an estimate of the maximum likely dependence of the results on the rather uncertain value of the virtual mass coefficient. I n practice we are concerned with fluidized beds bounded by the vertical walls of their containers, and a t these walls the normal components of fluid and particle velocities must vanish. T h e simplest Lvavelike solution consistent with such a boundary condition has lvave vector pointing vertically upward and horizontal \vave fronts. so that
2
(%)+ G)
+4(iy[F(A
- F)
-?]
lim a(k,) = bE
and
kz+
q = 4
(i)
(AE - F -
-
FBD c )
(33)
Taking the negative signs before the square roots in Equations 30 and 31 it is seen thdt E is always negative, so the corresponding voidage perturbation is damped out with increasing time. Taking the positive signs, on the other hand, [ may be positive and indeecl very frequently is, so in investigating the possibility of instabilities generating voidage fluctuations we may further confine attention to Equations 30 and 31 with the positive choice of signs. Equations 1’ to 4‘ and all subsequent developments correspond to the particular choice of the virtual mass term with
(35)
/.,--to
(32)
m
lim ~ ( k , )= k,-m
2bF A
(37)
From Equations 34 and 35 it is seen that 5 0: k X 2 and 7 a k, when k , + 0, so the velocity approaches a constant limiting value, ivhile Ax l / k , 2 and tends to infinity. Thus the growth rate tends to zero and the growth distance increases without bound when the wavelength becomes very long. Similarly, from Equations 36 and 37, approaches a constant value and 7 rn k , as k , + m , from which it follows that both the velocity of propagation and the growth distance approach finite limiting values for very short wavelengths. This should he contrasted with the results for a simpler model previously VOL. 7
NO. 1
FEBRUARY 1968
15
investigated by Jackson (1963), where the growth rate was found to increase without bound a t short wavelengths. T h e more rational behavior of the present model is a result of taking into account the stresses arising from particle-particle interaction. From Equations 30 and 31 f and a are seen to be functions of k , and k in general, so regarded as functions of the three components ( k z , k,: k,) they show considerable degeneracy. This can be used to construct solutions more general than the simple unmodulated plane wave, since a linear combination of solutions lvith different wave vectors still has a unique groivth rate and propagation velocity, provided all the solutions combined correspond to the same values of k , and ik . In particular it is useful to construct solutions with axial symmetry about a vertical axis, since these may be chosen to simulate various boundary conditions a t the vertical walls confining a bed of uniform circular cross section. If r denotes the radial coordinate of a set of cylindrical polar coordinates with vertical axis, it can be shown (Anderson, 1967) that el =
il
Jo(k,r)exp ( f t ) exp [i(k,x
-
at)]
(38)
is a solution of this type, where f and a are calculated from Equations 30 and 31 with k,2 k,2 = k,2. (This result can be obtained either by superposition of an infinite number of plane \raves \vith wave vectors lying on the surface of a cone rrith vertical axis, in which case the Bessel function, J,, arises in the form of Sommerfeld's integral, or by relvriting Equation 19 in cylindrical coordinates and separating variables.) Equation 38 once more represents a wave propagating vertically up\vard, but now the amplitude is modulated radially by the factor J,(k,r). k , may be chosen to satisfy boundary conditions imposed in the radial direction-for example, if the bed is confined within a tube of diameter d,, a very severe boundary condition is obtained by requiring that the particles should be immobilized a t the tube wall, so that c 1 = 0 a t T = d C / 2 . T h e simplest solution satisfying this condition is obtained by requiring that r = d 1 / 2 should correspond to the first zero of J ,
+
k,dJ2 = 2.405 or k, = 4.810/dC
+ c'd,
T h e propagation of a small disLurbance is characterized by its amplification factor, 5, and phase velocity, V,, as functions of the wave number, k. If attention is confined to axially symmetric disturbances, the radial component of k is fixed by radial boundary conditions, so 5 and V p may be regarded as functions of k,. They may be computed from Equations 30, 31, and 17 for given particles fluidized by a given fluid a t a given voidage, provided the value of certain physical properties of the bed are known-namely, ii,. n,Po'/Po, C, X,S 4/3p,9, and p,S'-and we first discuss the extent to which it is possible to measure or estimate these quantities. T h e literature contains accounts of many methods of estimating the fluidizing velocity, io,a t least for minimum fluidization conditions, but for the present purposes it is unnecessary to rely on these. since iocan be determined directly as a function of voidage by measuring the bed expansion with increasing fluid velocity for the systems studied. These measurements also permit the form of the function P(E) to be found with reasonable certainty, since, if \ve neglect departures from uniform fluidization, the fluidizing velocity and voidage are related by Equation 10 which gives
+
P(%)
=
(1
- €0) ( P s - P,)g
= EO(1
UO
- -(Ps - P/)g uo €0)
(41)
Figure 1 shows values of deduced in this way from experiments on the fluidization of glass beads of 0.086-cm. diameter with water. T h e Richardson-Zaki equation (Richardson and Zaki, 1954) leads to an explicit equation for P as a function of e,-namely,
where u t is the terminal velocity of free fall of a single particle and S c a n be correlated as a function of the Reynolds number, u , d , p f / p f . Hojvever, for the present purpose u t and A' can be regarded as parameters available to fit Equation 4 2 to the experimentally determined values of and Figure 1 shows the very satisfactory fit obtained in this way by taking u t = 12.8 cm. per second and *I ='3.0 for the case of the glass
(39)
In fact, it is unlikely that a smooth vertical wall will completely prevent all particle motion, but it will restrict it because of friction betkveen the particles and the wall. One simple way of taking account of such a restriction is to require that €1 should vanish, not at r = d,/2 but a t some larger value of 7 . Thus if we urite d, = d ,
Evaluation of Propagation Properties of Disturbances
6oo
r\ t \
(40)
where d , is the particle diameter, and require that k, = 4 . 8 1 0 / d U
(39 '1
in place of Equation 39, we obtain a solution in which € 1 vanishes a t a distance c'dp/2 beyond the retaining walls. When c ' -+ 0, Equation 39' reduces to 39 and when c' + 00 we obtain the uniform plane wave solution. Of course the solution need not necessarily have axial symmetry even when the bed is contained in a tube of circular cross section with vertical axis, but the axially symmetric solutions go one stage further than the uniform plane waves in permitting wall effects to be taken into account. 16
l&EC FUNDAMENTALS
.40
.45
.50
.55
.BO
.65
.;
& Figure 1, Experimental values of /3 and fitted RichardsonZaki equation for glass beads with d, = 0.086 cm. Curve represents fitted equation
beads with d , = 0.086 cm. Having fitted an equation of the form of Equation 42 it follows that
(43) giving the property required in the stability calculations. T h e correct value to take for the virtual mass coefficient, C,. in a situation as completed as a fluidized bed is not knoivn. For an isolated particle. in an infinite fluid C = 0.5 and this value will normally be adopted here, but calculations for other values of C, will also be carried out to illustrate the effect of C, on the propagation characteristics. Effective shear viscxities of gas-fluidized systems have been measured by a number of workers, some of the most recent and probably most reliable results being those obtained by Schugerl, Mertz, m d Fetting (1961) using a rotating cylinder viscometer. Corresponding measurements on iiquidfluidized beds have not been reported, but the results of a preliminary investigation (Anderson and Bryden, 1965), instigated by the present writers appear to give viscosities of the same order as for gas-fluidized systems. T h e observed behavior a t very low rates of shear is non-Keivtonion of the pseudoplastic type, but a t higher rates of shear the effective viscosity takes a constant value which depends on the particle size and the voidage, but is of the order of several poises. Anderson and Bryden's (1965) values of effective shear viscosity for the 0.086-cm. diameter glass beads are quoted in Table I. Since the change from air to water as a fluidizing fluid does not greatly change the measured viscosities, it is reasonable to associate these values entirely with the resistance to shear of the particle phase and hence to identify them with pas. T h e particulate bulk viscosity, A,', is, of course, a quantity much less amenable to experimental investigation and there is a t present no direct experimental evidence bearing on its value, although Murray (1965) has suggested on theoretical grounds that it may be large compared with the shear viscosity. I n view of this uncertainty we shall calculate propagation characteristics for a range of values of (A,' 4//3 pos) extending to four or five times the estimated value of the shear viscosity. O n the value of the final relevant property-namely, p,9'-there is a t present no real evidence, either theoretical or experimental, and we once again adopt the procedure of exploring a range of values to get some idea of its effect and importance. T h e range considered is based on a suggestion of Anderson (1967) that ~dp8/d€1,=,, may be of the same order of magnitude as the quantity
+
T h e calculations of 6 and V , fall naturally into three groups. T h e purpose of the first group (Figures 2 to 8) is to investigate the influence of the physical parameters noPo'/&,, C, X,b 4/3 p,9, and ,bos', together with the radial boundary conditions, on a given physical system a t a given voidage. Calculations of this type have been carried out for a number of systems and the ones presented here as typical refer to glass beads of 0.086-cm. diameter fluidized by water a t a voidage eo = 0.46. T h e purpose of the second group of calculations (Figures 9 and 10) is to investigate the effect of varying the fluidizing velocity, and hence the voidage, for a given system with given estimates of the physical parameters referred to above, and once again results are reported for the same system of glass beads fluidized by water. Finally the third group of calculations (Figures 11 and 12) extends the range of the \vork to systems of different physical properties and, in particular, to systems fluidized by gases rather than liquids. Here we present the propagation characteristics of the same glass beads fluidized by air, rather than water, to illustrate the very large effect of changing the solid-fluid density ratio. Figure 2, illustrating the effect of terms arising from the particle phase stress tensor, immediately shows the most important difference between the present work and the simplified treatment earlier presented by Jackson (1963). Curves 1 and 2, in which viscous effects in the particle phase are neglected, give an amplification factor which increases monotonically \vith k,, so that there is no dominant Ivavelength more strongly amplified than all others, but instead the amplification becomes very large a t short wavelengths. m as k, m , but the continuum theory Indeed formally 6
+
- -
06
-
0.5
-
p
0.4
cn
u
Ln 0.3
0.2
0.1
1 .o
0
Table 1.
io,
Eo
0.42 0.46 0.50
Cm./Sec., Exptl. 1.oo 1.27 1.61
nomP o ' l P o ,
won Poises, Exptl.
3.90 3.47 3.10
8
Fitted Equation
5 3
+
I+a/bI,,,
4 / 3 ~ 0 a DweslSq. Taken as Cm. (Esti4p0a mated)
32 20 12
0
3.0
Figure 2. Effect of particle phase stress tensor on arnplification factor
Physical Parameters for Glass Beads Fluidized by Water ( d p = 0.086 cm.) Xo'
2.0
16 18 20
Curve No. 1 2 3
Xoa
+
4/3M08,
Poises 0
0 10 10 20 20
4
5 6 For all curves eo =
IdPldelo, DyneslSq. Cm. 0 18 0 18 0 18
0.46, noPo'/Po = 3.47,Co = 0.5, k, = 0
VOL. 7
NO. 1
FEBRUARY 1 9 6 8
17
I
5 o r
1
1 4.0
3.0
2.0
Figure 2. A4nincrease in this quantity decreases the amplification factor for all values of k , and the effect can be large, but a finite value of p l ’ alone is not capable of inducing a maximum in the EGu,k, curve. k , = 0 for all results presented in Figure 2. Figure 3 shows the relatively minor effect on the phase 4!3 p,8. T h e velocity of introducing finite values of X,S system clearly becomes more strongly dispersive as this quantity increases. Changes in the value ofp,” are found to have a negligible effect on V,. Figures 4 and 5 illustrate the effect of changing the value of noPof/Po on the amplification factor and phase velocity, respectively. T h e change considered is considerably larger than the experimental uncertainty of this quantity and the effect is still small, so it is unlikely that uncertainties in the value of noPG’l/Po are important for the theory. Figure 6 shoxvs the effect of varying the virtual mass coefficient over the range 0 to 1.5, T h e amplification factor increases rapidly with C, for all values of k,: as might be expected, since the virtual mass effect may be regarded as contributing to the inertia of the particles. Although the form of the us. k, curve remains unchanged, it is clearly important to know the correct value of C, if quantitatively accurate predictions of amplification rates are needed. Changing C, was found to have a negligible effect on the phase velocity. Figures 7 and 8 illustrate the effect of requiring the radial wave number to take a finite value corresponding to some restriction of the vertical motion of the particles a t the boundary of the bed. T h e value k , = 1.89 cm.-‘ used in the calculations corresponds to d, = 2.54 cm. = 1 inch, so it simulates the severe restriction that the bed is confined Ivithin a tube of 1-inch internal diameter and the particle motion is completely inhibited a t the tube wall. ,4s might be expected, the wall effect introduces marked damping, so the amplification factor is reduced for all values of k , and the phase velocity is also reduced slightly. T h e effect on the propagation characteristics of changing the bed expansion is shown in Figures 9 and 10. For all these calculations C, = 0.5 and h-, = 1.13 cm.?, while the values
+
e
-
I
r
2 0
10
k, Figure 3. velocity
40
30
5 0
60
(ern-')
Effect of particle phase stress tensor on phase Curce
-t 4/3Poa
.Yo. 1 2
0 10 20
3 Value of Idp’/de’, did not affect V , signiflcantly. e, = 0.46, n,0,‘//30 = 3.47, C o = 0.5,k, = 0
For all curves
on lvhich the analysis is based ceases to have any meaning \vhen the \Tal-elength becomes comparable lvith the particle diameter. T h e greatest value of k , appearing on Figure 2namely, 7.0 cm.-‘-corresponds to a Ivavelength of a little less than 1 cm. and probably represents the limit of meaningful calculations. IVhen X,S 4/j3 p G s is given a finite value, on the other hand, the amplification factor is reduced a t all \vave numbers and it passes through a ivell defined maximum a t a finite value of k,: rvhich therefore determines a Xvavelength of maximum amplification. T h e disturbances in voidage \Till therefore be dominated by ivavelengths in the neighborhood of one particular value as they grow. T h e effect of interparticle pressures depends on the value of p l f or, equivalently, (dp’/de),, whose values are quoted in
+
0.20
4 0
5
2
0.: 5
3.10
0.0 5
t
1.0
k
1
0
10
20
30
5.0
4.0
6.0
1
0
1.0
1
1
2.0
1
1
3.0
k, Figure 4.
Effect of no&’/p, on amplification factor
Curce
noPo’
.To,
P O
3.50 3.35
1 2 For both curves e o = 0.46, C, = ldpe/dej, = 1 8 dynes/sq. cm., and
18
1
3.0
>- 2.0
//
\ \ \ Y
I
e. c
i
l&EC FUNDAMENTALS
0.5,Xon kr = 0
+
4/3fi08
= 20 poises,
Figure
5.
1
1
4.0
1
1
1
5.0
1
6.0
(crn-0
Effect of no/3,f//30on phase velocity
Curve No,
Po
1
3.50
3.35
2
+
4/3/108 For both curves e, = 0.46, Co = 0.5, ioa Idpe/del, = 1 8 dynes/sq. crn., and kr = 0
= 20 poises,
! 0.3
0.3 n
v
-
0)
b v)
s
0.2 0.2
w )u\
0.1
0.1
0.0 0.0
k -0.05
L 1.0
2.0
\ I
3.0
A 5.0
4.0
Curve No. 1
Figure 7. Effect fication factor
on
1.5
4 For all curves e, = 0.46, no&'/& = 3.47, XOs Idp8/del, = 18 dynes/sq. cm., and kr = 0
+
3
0 1.89 0
4
1.89
2
1 .o 0.5 0
3
For all curves eo = 4/3p08 = 2 0 poises
*/3poa = 20 poises,
5.0
6.0
of radial wave number k , on ampli-
Curve No. 1
co
2
5.0
C,
4.0
k , (cm-')
6.0
k, (ern") Figure 6. Effect of virtual mass coefficient amplification factor
3.0
2.0
1.0
0.46, n,B,'/t?,
0 0 18 18 = 3.47, C, = 0.5, and hoa
+
1
I
w
0.1
284 2'o
1
t 1.0
2.0
3.0
4.0
6.0
5.0
k, (cm-') Figure 8. velocity
0
k, (crn-l)
Effect of r'adial wave number k , on phase Figure factor
Curve
.VO.
9.
Effect of b e d expansion on amplification
Curve
1
0
0
No.
2 3 4
1.89
0 18
1 2 3
For all curves ea = 4/3p08 = 20 poises
0 1.89
18
0.46, nlopo'/(3, = 3.47, C, = 0.5, and hoe
+
€0
0.50 0.46 0.42
For all curves C, = 0.5 and in Table I
VOL. 7
k, = 1.1 3
NO. 1
ern.-'
Other properties as
FEBRUARY
1968
19
taken for other bed properties are as listed in Table I. Both the amplification factor and the phase velocity increase rapidly as the bed is expanded, and the dominant wavelength decreases a little. Finally, Figures 11 and 12 shoi+ the amplification factor and phase velocity for a bed of the same glass beads fluidized by air instead of water. All calculations irere carried out for e, = 0.42, k , = 1.13 cm.?, C, = 0.5, 'dps,'dec u = 20 dynes per sq. cm. and n,p,'//3, = 3.91. Three different values for XOs pOs were considered, as indicated on the diagrams. T h e general forms of the curves of { us. k , and V , us. k , are similar to those for the \vater-fluidized system presented in Figures 2 and 3, except that the phase velocity is much more strongly dependent on k , in the air-fluidized system. However, the most striking difference is found on comparing the
+
5.0
ordinate scales of the graphs of E us. k , (Figures 2 and l l ) , which differ from each other by a factor of 100. Since the plotted curves are roughly comparable in size, this means that disturbances grow roughly 100 times more quickly in the air-fluidized system than in the \rater-fluidized system. This effect of increasing the density ratio is typical (Anderson and Jackson, 1964), and indeed the prediction of the extreme instability of gas-fluidized beds relative to liquid beds is perhaps the most striking success of the fluid mechanical theory of fluidized beds at the present time. Calculations for the air-fluidized system were also carried out a t other values of no&'///3,, C,? k,, and ldps/'de),,and the effects of varying n , p 0 ' : l ~ , and k, were found to resemble closely those already presented for the !rater-fluidized system. Variations in C, had a very much smaller effect, however, in the air-fluidized system and this is to be expected in view of the much smaller density of air. Increasing the value of ldps,l'dc~,,had little effect on the propagation characteristics in the air-fluidized system until it reached a value of about 200 dynes per sq. cm. This is comparable with the value of
4.0
3.0
for the system, which Anderson (1967) has suggested as an indication of the order of magnitude to be expected for
2.0
0
1.0
2.0
6 .O
5.0
4.0
3.0
k, (ern-') Effect of b e d expansion on phase velocity
Figure 10.
Curve
'TO.
(a
0.50 0.46 0.42
1
2 3 = 0.5 and k, = 1.1 3 cm.-'
For all curves Co in Table I
Other properties as
I t is not difficult to trace a connection between Anderson's tentative estimate of this quantity and its role in the stability theory, though Anderson's suggestion was not originally based on dynamic stability considerations. T h e quantity pas' enters the expression for the propagation characteristics only as a factor in the quantity e! given by Equation 28. Scrutiny of Equations 30 to 33 shoivs, ho\vever, that e appears only in the expression F ( A - F ) - ADe/b2
30
-
LO
I)
Q
2?.3 0 'WI 20
10
I
1.0
0
I
I
2,O
I
I
3.0
k,
I
I
4.0
L
I
I
5.0
I
6.0
ivo.
1 2
3
l&EC FUNDAMENTALS
2.0
3.0
5.0
4.0
6.0
k, (cm-') koa i4/3P'o'> ,
Potses 8 16 32
For all curves e, = 0.42, n0Po'/Po = 3.91, Co = 0.5, ldpa/dcl, = 2 0 dyner/sq. cm., and k, = 1.1 3 cm.-'
20
1.0
(cm-')
Figure 1 1 . Amplification factor using air as a fluidizing medium Curue
0
Figure 12. medium
Phase velocity using air as a fluidizing
Curve d\'o.
1 2 3 For all curves e, = 0.42, no@.'/& = 3.91, 20 dynes/sq. cm., and k, = 1.1 3 cm.-'
A,"
+
,4/3w0s,
Poises
8 16 32
C, = 0.5, Idp'/del, =
a n d if p s / p ,
>> 1 and we: take C,
=
of uniform fluidization = --v(dp'..'dt), = f a = quantity defined by Equation 33 = radial coordinate = exponent of time dependence for perturbations = time = local mean fluid velocity vector = local mean fluid velocity in state of uniform fluidization - 'U, = E,u,, fluid velocity of a superficial hasis = perturbation in local mean fluid velocity = amplitude of perturbation tvave in u 1 = Cartesian components of u = velocity parameter in Richardson-Zaki equation = phase velocity of perturbation ivave = local mean particle velocity vector = perturbation in local mean particle velocity = amplitude of perturbation Tvave in V I = Cartesian components of v = quantity defined by Equation 32 = coordinate on axis pointing vertically upxvard = Cartesian components of position vector = gror\.th distance of perturbation = local mean particle pressure in state
0:this reduces approximately
to
A (1 -
g)
T h u s dps 'de will have a significant effect on the value of this expression, and hence on the propagation characteristics, only \\hen it is sufficiently large that De,'bz 1. Using Equations 24, 26, and 28, this gives
-
\\hich agrees \\ith Anderson's estimate after setting C, = 0. T h u s Anderson's estimate leads to a value of dps/de f o , which is large enough to have a significant efTect on the propagation characteristics of small disturbances: Whether or not this corresponds to the phlsical facts is, of course, another question, but it is certainly not justifiable to neglect interparticle pressure altogether. as suggested by Murray (1965), without further experimental evidence.
GREEKLETTERS Acknowledgment
O n e of us (T. B. A , ) received financial support from the Science Research Council for the period of this work.
P(E) P 9
Po
631
e
Nomenclature
€0
el
quantity defined by Equation 21 quantity defined by Equation 22 quantity defined by Equation 26 virtual mass coefficient virtual mass coefficient in state of uniform fluidization quantity defined by Equation 27 quantity defined by Equation 40 and determining radial boundary condition quantity defined by Equation 24 particle d k m e t e r diameter of tube containing cylindrical bed effective diameter defined by Equation 40 quantity defined by Equation 23 effective stress tensor for fluid phase effective stress tensor for particle phase quantity defined by Equation 28 quantity defined by Equation 25 local mean value of contribution to fluid-particle interaction force gravitational force per unit mass gravity force vector per unit mass unit vector in the up\vard vertical direction Bessel function of zeroth order wave vector of perturbation Cartesian components of k radial component of k exponent in Richardson-Zaki equation local mean value of number of particles per unit volume local mean value of fluid pressure local mean fluid pressure in state of uniform fluidization perturbation in local mean fluid pressure amplitude of perturbation wave i n p , local mean value of particle pressure
€1
o
x A(€)
= fluid-particle drag coefficient per unit bed volume = P(td = -v(d@ de),,,, = Kronecker delta
local mean voidage local mean voidage in state of uniform fluidization perturbation in local mean voidage = amplitude of perturbation lvave in € 1 = imaginary contribution to s (s = - io) = I\ avelength of perturbation \cave = effective bulk viscositv for fluid Dhase = = =
=
A(€")
= effective hulk viscosity for particle phase =
XqE,)
= effective shear viscosity for fluid phase
= P(E0) = effective shear viscosity for particle phase = d e o ) = volume
of one particle
= grotvth rate of perturbation wave (s = = density of fluid = density of solid
5 - io)
literature Cited
Anderson, X. B., Bryden, J. O., "Viscosity of a Liquid Fluidized Bed," Department of Chemical Engineering Report, University of Edinburgh, 1965. Anderson, T. B., thesis, University of Edinburgh, 1967. Anderson, T. B., Jackson, R., Chern. Eng. Sci. 19, 509 (1964). Anderson, T. B., Jackson, R., IND.ENG.CHEM.FUNDAMENTALS 6, 478 (1967a). Anderson, T. B., Jackson, R., IND.ENG.CHEM.FUKDAMENTALS 6, 527 (1967b). Chappelear, J. E., IND.ENG.CHEM.FUNDAMENTALS 5 , 576 (1966). Jackson, R., Trans. Inst. Chem. Engrs. 41, 13 (1963). Molerus, O., Chem. Ing. Tech. 39, 341 (1967). Murray, J. D., J. Fluid .2riech. 21, 465 (1965). Pigford, R. L., Baron, T., ISD. ENG.C m h f . FUNDAMENTALS 4, 81 (1965). Richardson, J. F., Zaki, LV. N., Trans. Inst. Chem. Engrs. 32, 35 (1954). Schugerl, K., Merz, M., Fetting, F., Chem. Eng. Sci.15, 1 (1961). RECEIVED for review September 11, 1967 ACCEPTED Sovember 9, 1967
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NO. 1
FEBRUARY 1968
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