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
unstable. However, the remaining quadratic factor may also have a root with posiiive real part, and the corresponding condition for instability is derived. T h e growth rate is found to be independent of the horizontal component of the wave vector of a general perturbation. Thus no mechanism cf the type suggested above for stabilizing perturbations exists in this model.
We find, by combining Equations 11 and 12 with 9 and 10 in turn, that
1 - E
E
and
Derivation of Secular Eiquation
We assume that Equations 19 through 21 of (7) define the perturbed motion of the bed. If rewritten, these equations are
- i k 2 p + g L k2Apf __ d-f
= div
Equations 13 and 14, which are linear combinations of 3 and 4, can be used to replace they components of 3 and 4 . We can then equate the determinant of the coefficients to zero. Since Equation 13 depends only on f , one factor of the determinant is
v
dt
1 We take
+ iinx + iny) V = V ~:xp(at + i7n.x + iny) f = f exp (at + imx + iny) p = p exp (at + imx + in)) v
=
v exp(at
(14)
(5)
This is exactly Equation 33 of ( I ) . I t is there shown that one root of Equation 15 has a positive real part; hence the state of uniform fluidization is unstable to a two-dimensional perturbation. From Equation 14 we see that another factor is (the coefficient of p )
-ik2
(6) (7)
(8)
where v , V, f , and p are the unknown amplitudes of the perturbations. If the motion is assumed to be two-dimensional, then the substitution of Equations 5 to 8 into 1 to 4 yields six siinultaneoiis linear ey uations in six unknowns. For these equations to be consistent, the determinant of the coefficients must be zerg,. If we choose k to be the wave vector of the perturbation (inversely proportional to the wavelength) with components m and n and take the scalar products of Equations 3, 4, 1, and 2 with k , we obtain
+
=
0
(16)
which is independent of a. T h e remaining factor is obtained from the x components of Equations 3 and 4 and 11 and 12. We find
im
in 0
0 im
in
Expanding the determinant we find
where we have introduced p*, the weighted density. T h e two solutions for a are
We see that regardless of the sign of the real part of the square root, a t least one root will have a negative real part. T h e other root will be positive if
and
--af = ik
.v
(12)
We have already seen that the two-dimensional perturbation is unstable. Therefore, no detailed numerical evaluation of Equation 20 has been attempted. We notice from Equations VOL. 5
NO.
4 NOVEMBER 1966
577
15 and 19 that the rate of growth of the instability is independent of rn, the horizontal component of the wave vector of the perturbation. I t can also be shown that nothing new is obtained if the perturbation is three instead of two dimensional. The only change in the results is that Equation 18 is a double factor of the secular determinant. The notation of (7) is followed here to facilitate comparison of results.
t
= time
v
= superficial fluid velocity = superficial solid velocity = constant representing fluid-solid resistance, equal to
V cy
L/4
E%
AP = E
PS
- PF
= fraction by volume of fluid in bed in steady state
pF = density of fluid p,T = density of solid particles p* = epF (1 - e)pS, weighted
+
density of phases
r$
= function of void fraction used in equations
A
=
gradient operator
Acknowledgment
The author thanks T. Baron, who suggested this problem. Nomenclature a = exponential growth rate factor g L = local value of acceleration of gravity i = imaginary unit j = unit vector along y axis (vertical) k = wave vector, components rn and n m = wave number along .x axis n = wave number along y axis p = deviation of fluid pressure from steady state q = steady state superficial velocity along y axis
literature Cited
(1) Pigford, R. L., Baron, T.,IND.ENG.CmM. FUNDAMENTALS 4, 81 (1965).
J. E. CHAPPELEAR
Shell Development Co. Houston. ?'e.?, RECEIVED for review August 8, 1966 ACCEPTED September 6, 1966 Publication 450, Exploration and Production Research Division, Shell Development Co.
CORRESPONDENCE CONCENTRATION-DEPEN DENT PHYSICAL PROPERTIES AND RATES OF MASS TRANSPORT SIR: In recent correspondence between Rhodes (8) and King (2) there seemed to be general agreement that mass transfer coefficients will be concentration-dependent when physical properties are similarly so. T h e point a t issue appeared to be the relative importance of a variable diffusion coefficient. \Ye are also studying the effect of concentrationdependent physical properties on mass transfer for the dissolution of solid electrolytes into their solutions, and this letter presents preliminary data which are pertinent to the discussion. The apparatus and technique used in this work have been fully described (3, 5 ) . Briefly, a sphere of solute 1 1 4 inch in diameter fixed on a wire is suspended in a stream of solvent flowing upwards in a vertical tube 2 inches in diameter. Diffusion coefficients of electrolytes in \vater are extremely dependent upon concentration, and variations of i.1007, from values at infinite dilution to those near saturation are not uncommon (6). The systems which we have so far investigated are ammonium and potassium alums. For the former, the differential diffusion coefficients, Ddlfi, a t 20' C. range from 4.10 x 10-6 sq. cm. per second a t saturation [11.52 grams of N I - I ~ . A 1 ( S 0 4 )12H20 2. per 100 grams of HzO] to approximately 8.5 X 10-5 sq. cm. per second a t infinite dilution ( 4 ) . This represents a n increase of 1107,. Moreover, the change is not only large; it is also nonlinear, and accurate prediction of its effect is extremely difficult. Other physical properties of the solutions change over this concentration range but not to such an extent-for example, viscosity decreases linearly by approximately 30%, while density falls by approximately 4%. The physical properties of potassium alum solutions are very similar to those of ammonium alum (4). For these two electrolyte systems, diffusivity is obviously the most important concentration-dependent physical property requiring consideration. Some of our experimental mass 578
l&EC FUNDAMENTALS
30r 9 v
I
30
20
I
I
i l l 1
I
I
100
500
200
1000
Re Figure 1. Dissolution of ammonium alum in aqueous solution a t 20" C. Driving forces,
AC, grams hydrate/lOO
grams water
A 11 -52 0
5.92 1.06
transfer data are plotted in Figure 1, A and B, using the standard dimensionless groups. Figure 1A uses bulk solution physical properties, except for diffusivity, which is taken as that a t the interfacial concentration, as suggested by Ranz (7). Each line represents a different concentration driving
