Nomenclature
L
D
1 2
= diameter of tube, cm. =
I,
binary diffusivii.y. sq. cm./sec.
121
= length of tube: cm. = average moleciilar Tveight
:tf.,
=
literature Cited ( 1 ) Ananthakrishnan, V.: Gill, \V. N., Barduhn, A. J., A.Z.CI1.E. J .
molecular \\eight of A
.VA = flux of A Ivith r s p e c t to fixed axes? gram molel’sq. cm.
Yec.
i&,Dl D~~ radiiis of tube. cm. distance from center of tube, cm.
Pe
= Peclet number.
R r
= =
.I
=z,!’R
Phenomena,” p. 524, !$ley,
= mole fraction of
=r,R
i
= distance from bottom of tube, cm.
pd
= =
p
New York, 1960.
Fortran Programming,” \Viley, New York, 1964. (8) Singh, S.N., Appl. S i . Res. A7, 325 (1958). ( 9 ) Taylor, G. I . : Proc. Roy. Soc. ( L o n d o n ) AZ19, 186 (1953).
A
,k.4
11, 1063 (1965). ( 2 ) Aris, K.. Proc. Roy. SOC. (London) A235, 67 (1956). ( 3 ) Bird, K. B., Stewart. i2.’. E., Lightfoot, E. N., “Transport ( 4 ) Heinzelmann, F. J., LVasan, D. T., \Vilke, C. R., IND.ENG. CHEW. FCNDAHENTALS 4, 55 (1965). ( 3 ) Nichols, D. G., Lamb, D. E.: private communication, 1965. ( 6 ) Langhaar, H. L.: Trans. A.S.M.E. A64, 55 (1942). ( 7 ) iMcCracken, D. D., Dorn, \V. S.,“Numerical Methods and
7‘ - 3.4, 3no 7‘“ = value of 7‘ from Equation 7 u = mass average velocity, cm.jsec. i = molal average vrlocity. cm.,’sec. j
= value a t z = L = value based on one-dimensional model = value based on tlvo-dimensional model
S. S. RAO C. 0. BENNETT Liziversitj of Connecticut Storrs, Conn.
conccntration of A ? gram mole/cc. density of gas, $;ram molejcc.
RECEIVED for review March 31, 1966 ACCEPrED August 25, 1966 Computational part of this work carried out in the computer center of The University of Connecticut, using a digal computer. Supported in part by grant GP-1819 of the National Science Foundation.
SUBSCKIPTS b = bulk or average max = maximum 0 = value a t i = 0
MECHANI!SMS FOR ELBOW= AND DRUM-TYPE
MIXERS irq MIXING
O F SOLIDS
Elbow- and horizontal drum-type mixers were designed with comparable geometry, making possible direct comparison of mixing mechanisms. principal mechanisms of mixing have been proposed : convective: shear, and diffusive. Although some investigators have engagrd in speculation regarding the predominant mechanisms for different types of mixers? little substantiating evidence ha; been reported and the designation of mixing mechanisms has remained largely intuitive. Differences in geometry. and in particular the size, shape, and location of the access ports, have made it impossible to use comparable sampling techniques for different mixers; this, in turn. has made it difficult to obtain comparable kinetic data. ‘To circumvent these difficulties and obtain preliminary comparisons of mixing mechanisms, elbow and horizontal drumtype mixers were designed and constructed in such a manner that the same sampling gyide and thief could be used for each. Also, those portions of each mixer in which the mixture resided a t the time of sampling ivere made comparable \vith respect to geometry. T h e results obtained with these mixers should therefore be particularly suitable for direct comparisons of mixing mechanisms.
COMPONFNT
THREE
+A+B
LOADING PATTFRN
+ A 4
.DRUM MIXER
Experimental
Except for the design and construction of the mixers. the details of the sampling and testing procedures ivere substantially identical to those described previously ( 7 ) . Figure 1 sho\vs the important characteristics of the t\vo mixers. T h e sampling guide and thief (not shoivn) were similar to those drscribed previously. The drum-type mixer \vas simply a box, rigged so that it could be rotated smooth.ly about its longest axis. O n e entire side was removable for sampling. T h e elbow mixer differed
ELBOW M/K€/?
Figure 1.
Important characteristics of two mixers VOL. 5
NO. 4 N O V E M B E R 1 9 6 6
575
from the usual design in that the legs conhisted of rectangular troughs rather than round pipeb. and each leg was fitted with a removable sampling section which corresponded geometrically to exactly one half of the drum-type mixer. Discussion of Results
In a previous report ( 2 ) , segregation indexes, based on the statistic “chi square,‘’ \\.ere used in kinetic studies of the mixing process and \\ere defined a5 follo\i s :
W X
0
z
z
2 U
W CI W
rn LL
where S
0
= a
numerical index which indicates degree segregation of the mixture X2u = observed chi square for any mixture PI = expected chi square for random niixture X 2 s = expected chi square for segregated mixture
of
Preliminary plots (not shown) of segregation indexes for component mixtures indicated that the rates of mixing differed tremendously. the numbers of revolutions corresponding to one half life (segregation index, s, equal to 0.5) being 640 for the drum-type mixer and approximately 2 for the elbo\v-type mixer. Because the objective of this iriveqtigation \vas to determine differences in mechanisms rather than rates? the results were expressed in half-life units (revolutions divided by 640) prior to plotting. Figure 2 summarizes the results plotted in accordance v i t h second-order kinetic behavior.
J
a
0
a
9
0 W
e
MIXING
Figure 2.
TIME, HALF-LIFE
UNITS
Comparison of mixing rates
tion of any differences in the mechanisms of mixing for the two mixers studied. Although the mixers were selected because of their marked differences in physical characteristics, no differences in mechanism \Yere detected, suggesting that many mixers have similar mechanisms.
Conclusions
Results for each mixer exhibited a trend toward increased mixing rates as mixing proceeded. This is in accord with results of previous investigations ( 7 ) . The results exhibited a slight amount of scatter. HoLvever, in view of the very small number of revolutions corresponding to one half life for the elbow mixer, and other sources of error, deviations from the single curve sho\zn probably are ivithin the limits of normal experimental variations. The general shape and trend of the results for each mixer were entirely similar. Therefore, these data afford no indica-
literature Cited
(1) Gnyle, J. B., Gary, J. H., Zrzd. Eng. Chem. 50, 519-20 (1960). ( 2 ) Gnvle, J. B., Lacey, 0. L., Gary, J. H., Ibzd., 50, 1279-82 (1958).
JOHN B. GAYLEI Bureau o/ Mines l u s ~ u l o o s aAla. , Present address, National Aeronautics and Space .4dministration, Kennedy Space Center, Fla. RECEIVED for review May 13, 1966 ACCEPTED August 22, 1966
HYDRODYNAMIC STABILITY OF A FLUIDIZED BED The two-fluid model of Pigford and Baron predicts instability of a fluidized b e d when subjected to a general perturbation, since one of the factors of the secular equation i s exactly that found b y Pigford and Baron for a one-dimensional vertical disturbance. The growth rate of the instability i s independent of the horizontal component of the wave vector of the perturbation. I G F O K D and Baron ( I ) , using a tizo-fluid model of a fluidPized bed, showed that the state of uniform fluidization was unstable to vertical perturbations. No wavelength of maximum growth rate was found, since the grolvth \\‘asa n increasing function of the wave vector. I t was recognized there that viscosity provided a stabilizing influence. This report presents the analyais of the stability of a uniform State of fluidization to a two-dimensional perturbation using the t\vo-fluid model of ( 7 ) . T h e analysis was undertaken to see if the inertia of the mobile solid phase would stabilize a general perturbation for short wavelengths. I t was felt
576
I&EC F U N D A M E N T A L S
intuitively that this stability might result because the solid particles ivould be “reluctant” to change directions rapidly. This intuitive notion was shoivn to be incorrect for this model. HoLvever, some other unexpected results were found. The conclusions are : A general perturbation leads to a fourth-degree algebraic equation in a, the dimensionless growth rate. The equation has t\vo quadratic factors. One is identical with that derived in ( 7 ) in the analysis of stability to vertical perturbations. Since, as is shown there, one of the roots of this factor must have a positive real part, the general perturbation is always
