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JOHN B. GAYLE and JAMES H. GARY' U. S. Department of the Interior, Bureau of Mines, Tuscaloosa, Ala.
Mixing of Solids
with Horizontal Drum-Type Mixers Mixers were made from sections of standard tubing or pipe which were machined to give a round drum with minimum roughness at the junction
T H E O R B T I C A L L Y , A system of solid particles is completely mixed when the particles are randomly distributed. For practical purposes, however, such a system is completely mixed when further mixing produces no change in either quality or quantity of the final product or does not increase production costs. Actually, for many commercial operations, theoretically complete mixing seems undesirable. Three principal mechanisms of mixing have been proposed (Z)-i.e., convective, shear, and diffusive. I n the work reported here, horizontal drumtype mixers were selected so that one type of mixing (diffusive) predominated. Mixing rates appear to exhibit second order kinetic behavior over most of the mixing range and vary directly as the 1.45 power of drum diameter and inversely as the square of drum length.
Experimental
T o obtain slow diffusive mixing, horizontal drum-type mixers were used. I n this type of equipment, mixing takes place slowly (IS), and where an index, C, calculated from the following empirical equation, is less than 54 (3) it also takes place exclusively by diffusion : C =
ND0.47 X 0 . 1 4
(1)
where N is speed in r.p.m. ; D, inner diameter of cyclinder in meters; and X, fraction in per cent of mixer volume occupied by batch. Each mixer was made by roughcutting top and bottom sections from different pieces of standard tubing or pipe and machining to give a round drum with minimum roughness a t the junction between the sections. T h e drum was rotated by a set of motor1 Present address, Department of Petroleum-Refining Engineering, Colorado School of Mines, Golden, Colo.
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driven rollers provided with a mechanical counter and cutoff switch. As in the previous investigation (7), 14 X 20 mesh roofing granules, having similar physical characteristics but different colors, were used. Before charging, semicircular cardboard or sheet metal partitions were placed in the drum crosswise to divide the chamber length into sections. Thus, for an equally distributed four-component mixture, three partitions were inserted into the mixer to divide it into four equal sections and equal quantities of the four components added, one to each compartment. After leveling, the partitions were removed, and the top was secured. By developing linear rate relations between time and extent of mixing, a given set of data can be represented by a single numerical parameter. Also, some indication of the mechanism of mixing should be obtained. However, preliminary results such as those presented in the earlier report, were not compatible with the usual rate equations over any considerable range of data. I n the previous investigation, sampling points, deliberately located to preclude samples taken a t an interface, caused an appreciable lag between actual start of mixing and apparent start. Therefore, sampling points for the present investigation were located so that overlapping samples were taken along the entire length of the mixer. Fach successive sample position was displaced sidewise in order that removal of one sample would not affect the particle distribution in the adjacent sample. Only one set of samples was taken from a given lot of material, and a different mixer charge was used to determine the extent of mixing after each designated number of revolutions to preclude effects of mixing during sampling. I n this revised procedure the number of samples had to be limited by taking
them in a single horizontal plane approximately l / ~inch below the surface of the charge. Because radial mixing in a rotating horizontal cylinder is extremely rapid (4,it was assumed that mixing was radially uniform and that the sampling procedure would provide data substantially representative of the charge as a whole. This was verified by visual observations and by particle counts for samples taken from two levels. Although mixing rates were not identical at all levels, differences were small and did not significantly alter shapes or slopes of the plots. Accordingly, the samples were taken at a single level. Each of the point samples, consisting of approximately 40 particles, was drawn out with a spatula into columns of single particles, and distribution of components for the first 32 particles of each column was recorded. Values of chi square were calculated ( 7 ) directly for the four-component system. By combining counts for the various components in all possible ways, chi square values were also obtained for four threecomponent and five two-component systems. These values were used to calculate segregation indexes from
s = (x: - X 3 / ( X ? - x 3
(2)
where S is a numerical index indicating degree of segregation; x:, observed chi square for any mixture; expected chi square for random mixture; and x:, observed chi square for segregated mixture.
x;,
Results
Because particles of different colors were in contact at the interfaces, the revised method of sampling indicated that complete segregation does not exist even at zero mixing time; the observed chi square values ranged from 50 to 97% of those calculated for complete segregation, depending on number of VOL. 52, NO. 6
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JUNE 1960
519
Data obtained from various mixers and mixtures.
Each point represents counts for at least 3000 particles
Mixer length and diameter, respectively, in inches: A, 1 1 .OO and 7.03; 0, 1 1:OO and 5.12; 11.00 and 3.56; V, 11.00 and 2.08; X 5 . 5 0 and 3.56; 0, 2.75 and 3.56
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interfaces and length of the mixer relative to the diameter of the probe. Thus the lower values of the initial chi squares were for the short fourcomponent mixer while the larger values were for the long two-component mixer. Inasmuch as the segregation indexes should take into account the actual initial state of the mixture, the observed chi square values for zero mixing time (xi) were used in Equation 2 in place of the theoretical values used in the previous investigation. Values for S,vary from unity to zero as mixing proceeds from complete segregation to complete randomization and were used to establish a conventional rate equation relating the rate of change to the degree of segregation :
- dS/dR = k ( S ) n (3) where dS is change in degree of segregation; dR, change in number of revolutions of mixer; k , rate constant; S, degree of segregation; and n, constant indicating order of the process. Most previous investigators have considered mixing as a first-order process and consequently have integrated Equation 3 for n equals one. However, first-order behavior usually is approximated only over relatively small ranges of data. Because similar results were obtained in this investigation Equation 3 was integrated for n = 2, and the process was tested for second-order behavior by plotting 1/S us. r.p.m. Such plots yielded straight lines over most (90%) of the range, but as randomization was approached, the points sometimes tended to deviate in a positive direction from a straight line; this indicated a higher mixing rate as the particles became more randomly dispersed. I t has been pointed out (2) that a change in rate was to be expected in the final stages of mixing as a result of the diffusion analogy. However, a slower rate was predicted, and the opposite was actually encountered. This could reflect a change in the mechanism of mixing, or some variation in the 520
physical conditions within the drume.g., a variation in the static charge or the influence of tiny fragments abraded from the particles-but more probably it reflects a characteristic of the plotting method. Thus for any plot of reciprocal concentration us. time or its equivalent, the skew nature of the distribution function makes large positive rather than large negative deviations from the line more probable; in the present instance, the possibility of negative S values (corresponding to oriented systems) would be expected to exaggerate this effect. By making additional plots of the data using an average mixing rate for each of the various mixers, it was found that slopes of the second-order plots varied as the diameter raised to the 1.45 power and were consistent with Lacey’s (2) prediction that mixing rates should vary inversely as the square of the mixer length. Using these relations, value of k in Equation 3 was partitioned to give a new equation showing that mixing rate depends on mixer diameter and length
- dS/dR = k ’ D 1 W 2 / L 2 (4) where D and L is inside diameter and length in inches, of the mixing d r u m ; and R, number of revolutions. Collecting like terms and integrating Equation 4 with respect to revolutions, 1/S = 1 ktD‘.45R/L2 (5) I n preparing the illustrated plots, approximately one third of the points had to be eliminated to reduce crowding of the graphs. However, eliminated
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Slow Diffusive Mixing Was Ensured Inside Dimens., Fraction Rotation In. Loaded, Speed. Length Diam. % R.P.M.“ 11.00 11.00 11.00 11.00 5.50 2.75
7.03 5.12 3.56 2.08 3.56 3.56
40.0 35.0 39.0 47.5 39.0 39.0
20.4 28.0 40.4 69.1 40.4 40.4
C 15 18 22 30 22 22
Inside peripheral speed held constant at 37.6 ft./min. for all mixers.
INDUSTRIAL AND ENGINEERING CHEMISTRY
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points were exclusively those falling on or near the lines, and as a result, the graphs reflect a degree of scatter somewhat greater than that actually observed. For each mixture, data for the different mixers are compatible with a single linear curve and thus indicates the adequacy of the corrections for drum length and diameter. The tendency for two sets of data to deviate from linear behavior in the direction of faster mixing as mixing nears completion has already been discussed. T h e number and distribution of components influences the rate at which mixing progresses-e.g., slopes of the upper two graphs on the left of the illustrated plots, representing systems having only oneinterface each, are appreciablysmaller than for the remaining three graphsrepresenting systems having two and three interfaces each. This indicates that rate of mixing for a given number of components increases with the number of interfaces. Similarly, variations in the number of components influence the rate of mixing even when the number of interfaces remains constant. Finally, mixing rate seems to depend somewhat on the relative proportions of the various components. Although it is desirable to partition the k value in Equation 5 further to take these trends into account, this has not yet been accomplished. References (1) Gayle, J. B., Lacey, 0. L., Gary, J. H. IND.ENG.CHEM. 50,1279-82 (1958). (2) Lacey, P. M. C., J. Appl. Chem. 4, 257-67 (May 1954). (3) Oyama, Y., Bull. Inst. Phys. Chem. Research ( T o k y o ) 12, 953 (1933). (4) Oyama, Y., Ayaki, K., Kugaku Kikui 20, 6 (1956). (5) Weidenbaum, S. S., “Advances in Chemical Engineering,” vol. 11, p. 291, Academic Press, New York, 1958. ( 6 ) Weidenbaum, S. S., Bonilla, C. F., Chem. Eng. Progr. 51, 275-365 (January 1955). RECEIVED for review September 14, 1959 ACCEPTED March 17, 1960 Work carried out in cooperation with the University of Alabama.
