definite removal of the constant temperature constraint in these methods should increase one’s confidence in sorting out less reliable vapor-liquid equilibrium data. These proposed methods are more sensitive to experimental errors than those based on the Gibbs-Duhem equation. Therefore, if errors are large, more data points must be available, so that these recommended methods can be successfully applied. Figure 1 shows the use of the similarity concept to estimate 6, from which the phase enthalpy differences of the given system are found. From these, a consistency test of data by the new method was made. An analogous approach of using the hexane-benzene system in the Gibbs-Duhem equation cannot produce the estimated heats of mixing for the pentane-benzene system in a consistency test. In Figure 2, a constant 6 value estimated from two data points enabled a good consistency test for data points in the same local region containing the tjvo data points. This estimated 6 definitely contains the bias of these two data points and appears less desirable than the use of a similarity approach. However, if thermodynamic consistency can be viewed as the internal consistency a m m g all data points, the test procedure as illustrated is acceptable. Even though the exact size of the composition region within which a constant 6 may be used varies from system to system, the simplicity and convenience of such a n approach for a multicomponent system are overwhelming advantages. They are apparent especially when data of similar systems are not available. For high pressure systems, the success of these proposed methods depends on the feasibility of the estimation of 6, which becomes a dominant fraction of a small AH a t high pressure. The similarity approach of using a constant 6 for a local region would be a safer procedure. JYith this, the method illustrated in Figure 2 should be applicable to high pressure systems, if fugacities are used instead of assuming fugacity coefficients as unity. Conclusions
An intrinsic advantage at the expense of more computation labor was observed : the by-product of providing the estimated phase enthalpy differences for a given system while the data
are to be tested. These enthalpy differences are the usually unavailable information needed for the design of separation equipment. Nomenclature
f
= fugacity, atm.
H = enthalpy, calories per gram mole Q = dimensionless excess free energy, Tao (1962) R = gas constant T = absolute temperature, OK. x = mole fraction of a component in liquid y = mole fraction of a component in vapor A = difference 6 = defined by Equation 2
SUPERSCRIPTS
*
= ideal gas = purecomponent
SUBSCRIPT
i = identity of a component Literature Cited
Hougen, 0. A . , Watson, K. M., Ragatz, R. A., “Chemical Process Principles,” Part 11, p. 1009, Wiley, New York, 1959. Ibl, N. V., Dodge, B. F., Chem. Eng. Sci.2, 120 (1953). Myers, H. S., Ind. Eng. Chem. 47,2215 (1955). Perry, J. H., Ed., “Chemical Engineers’ Handbook,” 3rd ed., p. 215, McGraw-Hill, New York, 1950. Rossini, F. D., Pitzer, K. S.,Taylor, W. J., Ebert, J. P., Kilpatrick, J. E., Beckett, C. W., Williams, M. G., Werner, H. G., “Selected Values of Properties of Hydrocarbons,” Natl. Bur. Standards, C461, pp. 239, 240 (1947). Sinor, J. E., LYeber, J. H., J . Chem. Eng. Data 5,243 (1960). Tao, L. C., American Institute of Chemical Engineers Meeting, Tampa, Fla., Paper 16D, 1968. Tao, L. C., Znd. Eng. Chem. 56, No. 2, 36 (1964). 1, 119 (1962). Tao, L. C., IND.ENC.CHEM.FUNDAMENTALS Van Ness, H. C., “Classical Thermodynamics of Nonelectrolyte Solutions,” p. 147, Pergamon Press, New York, 1964. RECEIVED for review April 10, 1968 ACCEPTED.August 1, 1968
A FLUID MECHANICAL DESCRIPTION
OF FLUIDIZED BEDS Comparison o f Theory and Experiment T. B. ANDERSON AND R O Y JACKSON’ University of Edinburgh, Edinburgh, Scotland
N A
previous publication (Anderson and Jackson, 1967) equa-
I tions of continuity and motion were developed to describe the behavior of fluidized systems. The present paper describes an experimental investigation to test the validity of these equations. At present, solu-tions of the equations are known for only two physical situations-the motion of a n isolated bubble in a fluidized bed (Jackson, 1963b; Murray, 1965b), and the 1 Present address, Department of Chemical Engineering, Rice University, Houston, Tex. 77001
propagation of small perturbations about the state of uniform fluidization (Anderson and Jackson, 1968 ; Jackson, 1963a; Murray, 1965a; Pigford and Baron, 1965)-so any experimental investigation must be confined to one or the other of these cases. Bubble motion has been the subject of a number of experimental investigations in recent years and one of the most striking theoretical predictions-the existence of circulating clouds of gas in the neighborhood of sufficiently large bubbleshas been elegantly confirmed by the use of colored tracer gas VOL. 8
NO. 1
FEBRUARY 1 9 6 9
137
Previously published equations of motion for a fluidized bed predict that the state of uniform fluidization is unstable, and that the instability takes the form of rising and growing fluctuations in voidage. In liquid fluidized beds the growth of the disturbances i s sufficiently slow to permit measurement of their propagation properties, and comparison of the results of these measurements with the predictions of stability theory provides an experimental test o f the proposed equations of motion. In view of the limited accuracy of the experimental technique and the sparseness of available data on bed properties, the agreement found is encouragingly good.
(Rowe and Partridge, 1963 ; Rowe et al., 1964). Comparison of experimental cloud dimensions with the theoretical predictionsofDavidson (1961), Jackson (1963b), and Murray (1965b) shows that the more complete solutions of Jackson and Murray are in better agreement with experiment than Davidson’s earlier and simpler results. Davidson and Murray base their treatments on the approximation of uniform voidage in the dense phase and obtain solutions in closed form, while Jackson, who does not make this approximation, is forced to solve his equations numerically. However, he is thereby led to the interesting further conclusion that a bubble should be preceded by a mantle of dense phase in which the voidage is slightly higher than a t points remote from the bubble. Although the predicted effect is small and difficult to detect experimentally, Lockett and Harrison (1 967) recently claimed to have obtained qualitative evidence of its existence by instantaneous local capacitance measurements. Thus, some of the salient features of bubble motion predicted theoretically have been confirmed by experiment, and the agreement between theory and experiment is best for the theories which approach most closely to exact solutions of the equations of motion. Nevertheless, for two reasons, solutions describing bubble motion do not provide a very satisfactory basis for testing the validity of the equations of motion. First, all such solutions so far obtained stem from simplified forms of the equations in which stress tensor terms, other than the fluid pressure, are neglected. Consequently, comparison of these solutions with experiment gives a check only on the form of the basic moment u m balance and the fluid-particle drag force. Second, it has been suggested that certain aspects of bubble motion may be characteristic of flowing powders rather than suspensions (Zenz, 1967) ; this may well be so, particularly in regions below the equator of the bubble, though we believe that the fluid-mechanical description is appropriate around the upper surface. An experimental study of the propagation properties of small perturbations about the state of uniform fluidization is capable of providing much more detailed information about the significance of the various terms in the equations of motion. I t has been shown theoretically that the state of uniform fluidization is usually unstable, so that fluctuations in voidage should arise spontaneously and propagate upward through the bed, growing as they rise (Anderson and Jackson, 1968). In gas fluidized beds the predicted growth is so rapid that it would be very difficult to follow its course experimentally. However, in liquid fluidized beds, the growth is much slower and it is possible to measure the characteristic wavelength, speed of propagation, and rate of amplification of the spontaneous fluctuations. For these reasons the present work is devoted to an experimental study of the propagation properties of disturbances arising spontaneously in liquid fluidized beds, and a comparison of the results of these experiments with the theoretical predictions already available. 138
l&EC FUNDAMENTALS
Table I.
Ballotini Grade No. 7 No. 5 No. 3 2 mm.
Particles Studied Mean Diameter,
Cm .
0.064 0.086 0.127 0,207
Density, G./Cc.
2.95 2.86 2.86 2.70
Experimental Method
The systems studied were beds of glass beads fluidized by water, and fluctuations in voidage were detected by their effect on the transparency of the bed to a light beam. When a collimated light beam is passed through a bed of transparent particles, the fraction transmitted in the direction of incidence depends on the number of scattering surfaces encountered during its passage, and, hence, on the number of particles per unit volume, the particle size, and the thickness of the bed. T o retain a measurable intensity of transmitted light it is therefore necessary to work with beds whose thickness is not too great relative to the particle diameter. O n the other hand, the behavior of the bed may be dominated by wall effects if the bed diameter is too small a multiple of the particle diameter. Thus, there is a restricted range of bed diameters over which the light transmission technique can be applied successfully, and this represents a serious limitation, in addition to the obvious limitation to beds composed of transparent particles, The two most obvious alternatives to the light transmission technique are the instantaneous measurement of 7-ray transmission, and the instantaneous measurement of capacitance in an arrangement which uses a small region of the bed as the dielectric in a condenser. I n principle, the first is probably the most satisfactory technique available, but to obtain a count rate permitting sufficiently high frequency resolution the power of the 7-ray source required is prohibitively high for ordinary laboratory use. The second method suffers from the disadvantage that fringe fields in the neighborhood of the condenser plates prevent a closely localized voidage measurement. The particles studied in the present work, their mean diameters and densities, are listed in Table I. They are glass beads of the type known as “ballotini,” formed by a shot technique, and are neither perfectly uniform in size nor perfectly spherical in shape. The beads were fluidized by water in transparent acrylic tubes 12 feet in length and ranging from ‘/z to 11/2 inches in diameter, with sintered bronze disks as bed supports. The bed depth was several feet, except for certain calibration experiments described below. The acrylic tubes were mounted vertically inside a water-filled channel of rectangular cross section with two opposite faces of transparent acrylic sheet. This served as an optical box and ensured that, in the absence of beads, the light beam would retain collimation in passing
I
I
TRACE I , 2 i n ABOVE DISTRIBUTO T W C E 2, 6 in ABOVE DISTRIBUTOR
through the apparatus. A carriage mounted on the optical box supported the light transmission apparatus and could be traversed u p and down to permit measurements a t various levels in the bed. Frovision for lateral displacement of the light beam was also provided on the carriage, so that the beam could be adjusted to pass through the acrylic tube diametrically. The arrangement of the light transmission system is shown in Figure 1. The light source was a 50-watt projection lamp of the prefocus type, with a pinhole a t its focal point. Light passing through the pinhole was collimated by a condenser lens, and the diameter of the collimated beam was finally determined by a stop, The unscattered light penetrating the optical box and its contents entered a short, blackened tunnel of the same diameter as the beam, so as to eliminate the effects of stray light entering the apparatus as far as possible. A small diverging lens then spread the light entering the receiver over the sensitive surface of a photomultiplier, whose output was amplified and transmitted to an ultraviolet oscillograph to be recorded. Thse light transmitter and receiver were rigidly mounted on the traversing carriage in permanent alignment. Some care is needed in the choice of beam diameter. If this is so small that it is coinparable with the particle diameter, the signal contains a very large proportion of high frequency “noise” resulting from rapid scintillation of the transmitted beam as individual particles move in and out of its path. Increasing the beam diameter eliminates this difficulty, but if the diameter becomes comparable with the scale of voidage fluctuations in the bed, the system no longer has adequate spatial resolution. Ideall.~,a different beam diameter should be used with each different particle size, but trials indicated that a single beam diameter of 4 mm. was adequate for the present work, though a larger beam could have been used with advantage in work on the largest particles. Similar considerations impose a lower limit on the diameter of the bed itself. If this is too small a multiple of the particle diameter, the beam will occasionally pass through the bed without encountering any scattering surfaces, and there will be considerable fluctuations in the number of scattering surfaces encountered, giving rise to a very “noisy” oscillograph record. However, this is not a serious practical limitation, since beds as narrow as this should in any case be avoided because of the large influence of wall effects. There is also, of course, a n upper limit on the bed diameter imposed by the requirement that a measurable amount of light should be transmitted. Calibration of the oscillograph deflection in relation to bed voidage required care for two reasons. First, appreciable gradients in voidage inay lead to a n appreciable change in voidage over the length of a deep bed, so calibration should be carried out using a bed no more than a few inches deep. Measurement of the depth of a bed containing a known quantity of particles immediately ,gives the mean voidage, and in a short bed changes in voidage may be neglected, so this may be identified with a local voidage a t the point of traversal by the light beam. All voidage calibrations were carried out between 2 and Z1/2 inches above the distributor in beds approximately
TRACE 3, I 2 in ABOVE DISTRIBUTOR
I TRACE 4. B i n A B O k DISTRIBUTOR I
rRACE 7, 4 8 in ABOVE OISTRlBUTOR I T R A C E 8 . 6 0 i n ABOVE DISTRBUTOR
Figure 2. Oscillograph traces at various heights above distributor 0.1 27-cm. diameter beads in 1-inch internal diameter tube
3l/2 inches in depth. However, a second difficulty arose from the fact that the transmitting power of the water-filled acrylic tube and optical box varied appreciably with height even in the absence of any beads. This was not unexpected in an apparatus constructed from standard acrylic tubing and sheet, and was corrected by carrying out a series of light transmission measurements a t various heights with no beads in the tube. T h e signal a t each height was thus related to the signal a t the point where the voidage calibration was carried out, and subsequently, all experimental signals were corrected to equivalent signals a t this height before being translated into voidages. Calibration of the empty apparatus in this way was necessary before each experimental run, as its optical properties changed slowly, largely as a result of films deposited from the water onto the acrylic surfaces. After completing the calibration, the tube was charged with glass beads to a depth of several feet, the flow of water adjusted to establish the desired mean voidage, and oscillograph records of the photomultiplier output were taken a t a series of heights in the bed, starting just above the distributor. Well marked regions of high and low voidage could be seen to follow each other u p the bed, and their velocity of propagation was measured by timing their passage between fixed marks. Their dominant wavelength and rate of growth were estimated from the oscillograph records as described below. Complete details of the experimental procedure are given by Anderson (1967). Experimental Results
Figure 2 shows typical oscillograph traces obtained a t various heights above the distributor in a bed of glass beads of 0.127cm. mean diameter, fluidized in a tube of 1-inch internal diameVOL 8
NO, 1 F E B R U A R Y 1 9 6 9
139
I
0046
I
I
I
I
02 @ 004-
-
I
I
I
I
I
IN ABOVE DISTRIBUTOR
6 I N ABOVE DISTRIBUTOR
@
12 IN ABOVE DISTRIBUTOR
@ @
24 IN ABOVE DISTRIBUTOR
. -
36 I N ABOVE DISTRIBUTOR
0 03-
-
>
c Oi
-
W
-
z n
$
0
,
5
/
,
IO
15
,
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I UNADJUSTED
2o
o
A
0 2550 t
2 0 2 5 30 35 40 45 50 55 60 HT ( I N )
,
-
0
2 ADJUSTED
la-
016-
014-
0
u 5
IO
15 20 25 H T (IN 1
’ 1 0 0 30 35 40
I 0
I 5
15 20 25 30 HT (IN)
IO
Figure 4. Frequency and amplitude as functions of height above distributor from spectral analysis 0.127-cm. diameter beads in 1-inch internal diameter tube. 140
I&EC FUNDAMENTALS
e =
0.463
ter. The growth of a well defined periodic component on moving up the bed is plain, and correspondingly a visual inspection of the bed with backlighting shows it to be traversed by a regular sequence of light and dark bands corresponding to regions of high and low voidage. These stretch across the full width of the tube and move upward u i t h almost constant velocity, inducing periodic oscillations in the positions of the particles as they pass. A region of high voidage is sometimes overtaken by the next similar region below it, and the two coalesce, leading to a gradual increase in the mean spacing of the light and dense regions as the pattern moves u p the bed. The growth of a dominant disturbance shown by the oscillograph traces is also visibly distinguishable in most systems, as the appearance described above develops gradually on moving u p the bed; near the distributor there is little visual evidence of any density ivaves. Traces like those shown in Figure 2 are best analyzed by computing their power spectra, which will contain a peak corresponding to the dominant disturbance frequency. Spectra are obtained by Fourier transformation of autocorrelation functions computed from digitized chart records, but considerable care is needed in choosing the sampling interval for digitization, estimating the minimum length of record required, and carrying out the computations (Blackman and Tukey, 1960 ; Swinnerton-Dyer? 1962). I n the present work the dominant disturbances have frequencies around one to two cycles per second, and the nrork of Blackman and Tukey shows that a sampling interval of about 0.1 second and a trace duration of about 50 seconds are necessary to obtain a good representation of the spectrum over a frequency- range 0 4 5 cycles per second, which covers the frequencies of interest. Thus, a t least 500 points of each oscillograph trace must be digitized, and a complete spectral analysis for each of the traces recorded in this work represents an amount of data logging beyond the reasonable range of manual digitization. .4s automatic data logging equipment was not available, the majority of records in this work were therefore analyzed by a much simpler but less satisfactory method. In the alternative method the heights of 30 successive maxima and minima were read from the oscillograph trace, and the total time occupied by this segment of trace was also noted. The amplitude of the dominant disturbance component was then assumed to be equal to half the mean peak-to-peak amplitude obtained by taking differences between successive recorded heights, and its period was assumed to be the ratio of the total elapsed time to the number of recorded cycles. This type of analysis, though valid for a purely periodic fluctuation, is clearly unsatisfactory when a broad spectrum of frequencies is present in the record; indeed a certain amount of subjective judgment is needed in ignoring maxima and minima corresponding to very minor “shakes” in the record. The complete spectral analysis was carried out for only two experimental traces and the resulting spectra for one of these are shown in Figure 3, a t heights ranging from 2 to 36 inches above the distributor. Two inches above the distributor the spectral density decreases monotonically with increasing frequency and is typical of the spectra of random fluctuations. O n passing u p the bed, however, it is clear that a strongly dominant periodic component emerges and grows. The frequency and amplitude of this component are equal to the frequency and the square root of the amplitude of the spectral peak, respectively, and these quantities are plotted as functions of height above the distributor in Figure 4,A and B. The slow decrease in frequency with increasing height appears to result from the coalescence of disturbances traveling with slightly
Basis Fig: 4 Fig: 5
Table II. Properties of Dominant Component k , Cm.-‘ Ax, Cm. f,Cycles/Sec. 1.71 30.2 1.43 1.74 25.9 1.45
Cm:
A,
3.67 3.62
’
kYj--%-i0
40’ HT !IN )
0
0
o;
7b
60
-I
e o ‘
1
0071
0.174 0.203
0
2n n w
V,, Cm ./Set. 5.25 5.25
f , Set.-'
40
20
30 HT IN.)
IO
50
0 0 2 ~ S ’ . ” s
3
k
J i
c 3
n.
001
a
O
I
B o
2
P
i
2
a
13
/ :i I
1
0
W 0
20 30 40
5 0 60 70 80 90 100 HT ! I N )
8
30
1
1
40
Figure 6. Frequency and amplitude as functions of height above distributor
1
IO
1
20
HT ( I N )
0 02
0
‘
IO
0.064-cm. diameter beads in ‘/pinch diameter tube. e = 0.519 0
5
10
15
20
HT(IN1
Figure 5. Frequency and amplitude as functions of height above distributor from peak-to-peak analysis 0.1 27-cm. diameter beads in 1 -inch internal diameter tube.
internal
e = 0.463
different velocities, and since the coalescence of two similar disturbances produces a single disturbance of larger amplitude, part of the observed growth in amplitude must also be attributed to coalescence. If it is assumed that the amplitude varies in inverse ratio to the frequency for growth by coalescence alone, the observed amplitudes shown in curve 1 of Figure 4B should be reduced in the ratio of the observed frequency to the frequency a t points near the distributor. This yields the corrected amp11tudes given by curve 2 of Figure 4B, which is taken to represent the growth of the dominant component resulting from the instability described by Anderson and Jackson (1968) and others. Figure 4C, in which the corrected amplitudes are plotted on a logarithmic scale, indicates that the growth is approximately exponential over the first 12 inches of bed height, after which the rate falls off. Figure 5 again shows amplitudes and frequencies for the same system as Figure 4 , operated a t the same voidage, but in this case the plotted points were obtained by the simple “peak-to-peak’’ method of analysis described above. Their similarity with the results of Figure 4 is obvious, but a more precise comparison may be obtained by calculating the propagation properties of the disturbances from each set of curves. The properties of the dominant component of the voidage fluctuations which interest us are: T h e wavelength, A. T h e growth distance, A x , defined as the distance of travel in which the amplitude grows by a factor e . T h e velocity of propagation, V,.
The frequency, f. The wave number, k(k = 2 ./A). The growth rate, (, defined as the reciprocal of the time i n which the amplitude grows by a factor e . (Only three of these are independent-for example, k , 4, and V , determine A, Ax, and f.) The values of these properties obtained from Figures 4 and 5 are compared in Table 11. V, has the same value in both cases, as its method of measurement does not depend on the method of trace analysis. T h e values of wavelength and frequency obtained by the two methods are in good agreement, but the growth distance and growth rate differ substantially, though they are still of the same order of magnitude. Table I1 gives some indication of the shortcomings to be expected from the simple peak-to-peak method of trace analysis used in most of the work; in particular, the estimated growth distance and growth rate may be in error by about 20%. However, in view of the uncertainties of the theory and our sketchy knowledge of the correct values to take for physical parameters in the equations of motion, it may be regarded as very satisfactory if theory and experiment yield values of the growth rate which agree, even to this order of accuracy, over the whole experimental range. Thus, the inadequacies of the simple method of trace analysis are probably not serious a t the present stage of development of the subject. Figures 6 through 9 show the results of peak-to-peak analyses of a selection of oscillograph traces. T h e frequency is found to be a monotone decreasing function of height above the distributor, except in the case of Figure 8, where it passes through a minimum. I n all ca-ses the amplitude starts with a n interval of roughly exponential growth above the bed support; then the rate of growth falls off and the amplitude usually passes through a maximum and decreases again. I n only one case (Figure 9) VOL. 8
NO. 1
FEBRUARY 1969
141
ca
2 0 l
0 09
,
,
,
1
,
0 ,
,
I
I
l
I
1
t
I
1
1
~
~
1
0 08
w
2
0.06
ci
5a
0.04
0 02
P T (IN ) 0
Figure 7. Frequency and amplitude as functions of height above distributor 0.086-cm. diameter beads in '/yinch e = 0.504
20
IO
0
40
30
c
1
1
1
1
1
1
I
50
60
70
80
I
0.03
w
f
B
0.02
40 50 H T (IN.)
0.207-cm. diameter beads in 1 '/*-inch internal diameter tube. e = 0.451
0
0 2
20 30
Figure 9. Frequency and amplitude as functions of height above distributor
internal diameter tube.
H T (IN.)
0.04
IO
some circumstances. The occurrence of turbulence appears to be determined by some criterion of a Reynolds number type, since the turbulence disappears as the bed diameter is reduced or the fluidizing fluid velocity is decreased. On the other hand, the regular voidage waves can be observed as soon as the bed becomes fluidized, and are present even in beds of small diameter. Since the regular voidage waves and turbulent fluctuations may often be observed superimposed in the same system, it seems unlikely that they have a common origin. The origin of the turbulence possibly lies in the transverse modes found on factorizing the secular determinant of the linearized stability theory (Anderson and Jackson, 1968). Comparison of Theory and Experiment
a
3
a
001 L
4
I
,
1
0
IO
20
I
t
I
l
I
30 40 50 60 70
1
I
80
H T (IN.)
Figure 8. Frequency and amplitude as functions of height above distributor 0.086-cm. diameter beads in 1 -inch internal diameter tube.
e
= 0.485
does the amplitude increase monotonically and even then it may eventually pass through a maximum in a sufficiently deep bed. Of course, the results of the linearized stability theory refer only to the initial period of exponential growth and cannot explain the observed variation of amplitude higher u p the bed. Plots corresponding to Figures 6 through 9 are given by Anderson (1967) for all the experimental runs. The regular density waves just described are not the only observable disturbances of the steady state of uniform fluidization. At high voidages and in tubes of large diameter, a large scale turbulent motion of the particles is also present, and this turbulent motion and the regular voidage waves coexist in 142
I&EC FUNDAMENTALS
The propagation properties of the disturbances during their initial periods of exponential growth can be compared with the predictions of linearized stability theory (Anderson and Jackson, 1968), but for this purpose it is necessary to know the values of certain physical parameters of the bed:
+
(ion w o e ) , where Xo* and w o n are the effective bulk and shear viscosities, respectively, for the particle phase. Co, the virtual mass coefficient for a particle in the bed. pas', or ( d ~ / d n ) where ~ ~ , p * is the effective pressure of the particulate phase and n the number of particles per unit volume. no/30'/Po, where @ ( n )is the fluid-particle drag coefficient per unit bed volume, and Po' = (dp/dn).,. k,, the radial wave number for the disturbances, which takes account of the restrictions imposed on vertical motion of the particles by the walls of the containing tube.
The subscript zero indicates that each of these quantities is evaluated a t the voidage of the undisturbed bed. Our present knowledge of the values of these parameters is very inadequate and the extent to which it is possible to make reasonable estimates has been discussed by Anderson and Jackson (1968). In all the present calculations the virtual mass coefficient, Co, was taken to be 0.5 and the parameter
Table 111.
Exberimental
2-mm: ballotini
No. 3 ballotini
No. 5 ballotini
No. 7 ballotini
Observed Probagation
3.51 3.81 3.81 2.54 2.54 2.54 2.54 2.54 2.54 1.27 1.27 1.27
3.72 3.93 4.28 4.04 2.46 2.55 .2.72 .2.89 12.96 :2.53 :2.70 2.87
0,418 0,433 0.451 0,457 0,463 0,470 0,483 0.496 0,503 0.515 0.528
1.48 1.27 1.22 1.40 1.74 1.91 1.96 2.00 2.03 3.7 3.5 3.5
2.54 2.54 1.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27
'1.34 '1.47 '1.42 '1.64 l.77 0.86 0.98 3..02 1.10 1.20
0.469 0.485 0.504 0,529 0.541 0.495 0,507 0,511 0.519 0.530
2.57 2.36 3.63 3.55 3.55 4.31 4.25 3.98 4.06 4.13
0,500
0.151 0,273 0,398 ,
.,
0,203 0.264 0,422 0,496 , , ,
...
., , .,. 0.131 0.176 0.088 0.114 , , ,
0.051 0,104 0:128
...
Properties
Calculated Probagation
Parameter Values
6.62 8.01 8.95 5.84 5.25 5.36 5.87 6.10 6.48 3.92 4.42 4.75
0.90 1.10 1.20 1.25 1.70 1.90 2.10 2.20 2.20 2.10 2.30 2.50
0.li4 0,267
8.33 115 7.88 90 7.90 70 6.98 80 5.77 46.0 5.66 39.0 32.0 5.68 5.76 28.0 5.85 27.0 4. . 6._ 3 40.. -5 4.74 33.0 4.76 28.5
3.70 4.30 3.46 3.91 4.23 2.54 2.76 2.95 3.05 3.26
2.70 3.00 3.00 3.00 2.75 4.10 4.50 4.50 4.50 4.40
0.124 0.177 0.090 0.122 0.100 0.052 0.104 0.114 0.133 0.158
3.72 3.71 3.47 3.53 3.49 2.45 2.56 2.60 2.67 2.77
0.142 0.242 0.378 0.189 0.209 0.280 0.398 0.498 0.538 0.069
25.0 20.6 21.9 19.5 19.5 13.8 12.0 11.8 11.6 11.2
115 90 70 80 46.0 39.0 32.0 28.0 27.0
0.53 0.53 0.53 0.8 0.8 0.8 0.8 0.8 0.8
33.0 28.5
1.66 1.66
2.93 2.82 2.68 2.77 3.15 3.07 3.00 2.95 2.94 3.08 3.01 2.96
40 i
1 66
25.0 21.6 22.9 20.5 19.5 13.8 12.0 11.8 11.6 11.2
0.7 0.7 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66
3.39 3.38 3.30 3.08 2.96 3.55 3.43 3.38 3.31 3.22
noPo'/Po was determined from bed expansion measurements as described by Anderson and Jackson (1968). For the ballotiniwater systems studied, an idea of the order of magnitude of pa' can be obtained from the measurements of Anderson and Bryden (1965) and, following a suggestion of ,Murray (1965a), (XOs+"3 PO') was assumed to take a value four or five times as large as PO'. There is really no independent experimental evidence on the value of po". Anderson (1967) has discussed its estimation and, based on his suggestions, its value was taken to be comparable with ( X0s+4/3 PO') when both are expressed in c.g.s. units. Finally, a value of k , corresponding to a radial wavelength about three times the bed diameter was found to account reasonably well for the variation in propagation properties with bed diameter. I n comparing theory and experiment the observed dominant disturbance was assumlzd to correspond to the theoretical disturbance whose growth rate was a maximum with respect to wave number. Thus, the theoretical calculations were performed by fixing values of CO,k,, noPo'/Po, (X0s+4/3 PO'), and post, and maximizing 5 with respect to k. The values of CO,k,, and noPo'/Po were then regarded as firmly determined, but ( X O s S 4 / 3 10')and PO'' were varied, and the propagation properties of the dominant disturbance were calculated for a range of values of these parameters. T h e values of (XO' "3 PO') and po" finally selected then had to satisfy three criteria:
Table I11 compares the calculated properties of the disturbance of maximum growth rate with the observed properties of the dominant disturbance. The parameter values on which the theoretical figures are based are also quoted and are chosen as described above, placing most xveight on matching the observed and theoretical values of the groivth rate. I n some of the smaller diameter beds no experimental value is quoted for the growth rate. These are the cases mentioned earlier in which the oscillograph record is too noisy to permit a n accurate estimate of the groivth rate. Scrutiny of Table I11 shows that the agreement between observed and calculated values of the growth rate is good, as might be expected. The observed and calculated values of the propagation velocity, V,, agree Xvithin about 2070, while the observed and calculated values of the axial wave number, k,, do not differ by more than about 507,. The relatively large errors in the wave number are not unexpected as the maximum in the calculated us. k curve is very flat, so k , is not very well defined theoretically. The main features of the observed propagation properties are accounted for quite wellfor example, the growth rate is found experimentally to increase with increasing voidage for a given system, with increasing bed diameter, and with increasing particle size for a given voidage and bed diameter; the theoretical results bear this out.
1. They must not be inconsistent with available direct experimental measurements of their values. 2. They must be mutually consistent-for example, values of ( A 0 8 f 4//3 pas) at different voidages in a given bed must all lie on a smooth curve which decreases with increasing voidage. 3. Subject to the constraints imposed by 1 and 2, they should be chosen to give good agreement between theoretical and experimental propagation properties.
Discussion and Conclusions
+
Thus, there is a considerable element of fitting in selecting the parameter values, and at the present time the success of the theory must be judged by how closely it is possible to reproduce the observed propagation properties by calculations based on parameter values which are mutually consistent and not a t variance with our rather meager direct experimental knowledge.
The experiments may be criticized on the grounds that they were performed, of necessity, in beds of small diameter whose behavior may not be typical of larger beds. Two types of disturbances are well known to occur in narrow gas fluidized beds-round-nosed cylindrical bubbles, known as "slugs," and horizontal discontinuities, with a layer of closely packed bed above a completely void layer, from which it is separated by a horizontal interface. The velocity of propagation of slugs is related to the tube diameter, and particlesflow around them down the walls of the tube as they rise. The horizontal discontinuities, which can often be produced by a sudden reduction in flow of fluidizing fluid, rise slowly as particles detach from the interface, and run down through the void below it. VOL. 8
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I t is necessary to establish that the phenomenon observed in the present experiments is distinct from these. I t is clearly unrelated to the horizontal discontinuity just described, as nowhere is there a surface of discontinuity of voidage, nor does the voidage approach that of a packed bed even in the regions of lowest voidage, a t least in the earlier stages of development of the disturbances. However, the observed density fluctuation could result from the passage of a slug with its surrounding curtain of falling particles. There are four reasons why we believe that this is not the case. First, the appearance of the present disturbances in their earlier stages of development is unlike the appearance of slugs. Slugs have a characteristic and easily visible rounded nose, while the disturbances observed in this work appear to be almost uniform horizontaI bands stretching across the tube from side to side. Second, the observed velocities of propagation are much smaller than would be expected for slugs in tubes of the given diameter. Third, the rate of growth of the disturbances increases with increasing tube diameter, while slugs would be expected to develop most rapidly in the narrowest tubes, and fourth, the disturbances can also be observed as irregular rising stratifications in “two-dimensional” fluidized beds which are narrow in only one direction. The experiments have shown that an instability of the type predicted by the hydrodynamic theory can be observed in liquid fluidized beds, and that the quantitative values of the propagation properties of the resulting disturbances can be accounted for approximately by the theory. The accuracy of agreement between theory and experiment is not high in individual cases, but the general variation in observed properties from one experimental situation to another is correctly accounted for. I n view of the modest accuracy of the experiments and the method used to analyze their results, the un-
certainties in the values of certain parameters, and the early stage of development of the theory, it is feIt that the results provide encouraging evidence for the validity of the theory. Acknowledgment
The authors acknowledge the generous assistance of Imperial Chemical Industries, Ltd., which undertook the analysis of the two records for which complete spectra were obtained. One of us (T.B.A.) received financial support from the Science Research Council for the period of this work. Literature Cited Anderson, .4.B., Bryden, J. O., “Viscosity of a Liquid Fluidized Bed,” Dept. of Chemical Engineering Report, University of Edinburgh, 1965. Anderson, T . B., thesis, University of Edinburgh, 1967. Anderson, T. B., Jackson, R., IND.END. CHEM.FUNDAMENTALS 6 , 527 (1967). Anderson, T. B., Jackson, R., IND. ENC. CHEM.FUNDAMENTALS 7, 12 (1968). Blackman, P. B., Tukey, J. TV., “Measurement of Power Spectra from the Point of View of Communication Engineering,” Dover, h-ew York, 1960. Davidson, J. F., Trans. Inst. Chem. Engrs. 39,230 (1961). Jackson, R., Trans. Inst. Chem. Engrs. 41, 13 (1963a). Jackson, R.., Trans. Inst. Chem. Engrs. 41,22 (1963b). Lockett, M. J., Harrison, D., “Distribution of Voidage Fraction near Bubbles Rising in Gas Fluidized Beds,” Symposium on Fluidization. Eindhoven. 1967. Murray, J. D.,’J. FluidMech. 21, 465 (1965a). Murray, J . D., J. FluidMech. 22, 57 (1965b). Pigford, R. L., Baron, T., IND.ENG. CHEM.FUNDAMENTALS 4, 3 1 (1965). Rowe, P. N., Partridge, B. A., Chem. Eng. Sci. 18,511 (1963). Rowe, P. N., Partridge, B. A, Lyall, E., Chem. Eng. Sci. 19, 973 (1964). Swinnerton-Dyer, H. P. F., Computer J. 5, 16 (1962). Zenz, F. A., Hydrocarbon Proc. Petrol. Refiner 46, 171 (1967). RECEIVED for review AMay15, 1968 ACCEPTED November 6, 1968
AN EQUILIBRIUM THEORY OF T H E PARAMETRIC PUMP ROBERT L. PIGFORD, BURKE BAKER I l l , AND DWAlN E. BLUM Department of Chemical Engineering and Lawrence Radiation Laboratory, University of California, Berkeley, Calif. 94720
Very large separation factors have been obtained by Wilhelm and his co-workers using cycling flow of a binary mixture upward and downward through a column containing a fixed bed o f solid adsorbent which is alternately heated and cooled. The theory of such separations i s developed here on the assumption of local equilibrium between solid and fluid phases. The origin of the separation i s the ability of the solid phase to store solute deposited on it by fluid flowing from the bottom of the column and to release this solute later into another fluid stream which flows into the column from a top reservoir containing enriched mixture. The proposed mechanism takes into account the difference in the speeds of propagation of concentration waves through the packing during upward and downward flow.
HE “parametric pump” described by Wilhelm, Rice, and TBendelius (1966), Wilhelm, Rice, Rolke, and Sweed (1968), Wilhelm and Sweed (1968), and Wilhelm (1966) is a device which uses periodic fluid motion and periodic heat input to a tube filled with a fixed bed of solid adsorbent to separate the components of the fluid. Figure 1, taken from Wilhelm et al. (1968), shows how the separation apparatus is
144
l&EC FUNDAMENTALS
operated in the “direct mode.” Heat flows through the walls of the tube containing the fixed bed of solid absorbent particles. When the flow of heat is in phase with the changes in direction of the flow, as in Figure 2, there develops a net transport of matter along the flow direction which leads eventually, after several cycles, to a steady-state difference in the concentrations in reservoirs of fluid at the ends of the column. Thus,
