l o ,l H = ,I factors k = thermal conductivitv, B t u hr -ft -OF k I1 = mass transfer coefficient. f t hr .\ = impeller rotational >peed, r p s Xu = Nusselt number. h 7 k P = input pomer, ft Ib sec Pr = Prandtl number. C p p k = hear tranhferred. B t u Q Re = Re\nolds number, D2\p p sc = Schmidt number. p p D , Sh = Sheri\ood number. kIIT D T = tank dlameter, ft SL = sample location ratio = fluid bulk temperature, "F 7; = heater surface temperature. O F T, = ueight of sample after teht. g LV0 = \\eight of sample before test, g Wb
GREEKLETTERS a,
6, y
= constants
*
7l
*RPF
(1961).
= length nf run, hr.
0 0
= =
I.r
t
bafflle position. degree fluid viscositx, l b ft -sec
P
= @ @HEF = fluid densit\,
*
=
(4) Bennet, J. A. R., Lev,$ J . B., A.I.Ch.E. J.4, 418 (1958). (5) Brooks. G . , Su, G., Third National Heat Transfer Conference. .A.S.M.C.-.A.I.Ch.E., 1959. (6) Chilton, T. H.: Dreiv, T. B.. Jebens, R. H., Ind. Eng. Chern. 3 6 , 510 (1944). (7) Cuinmings. G. H.. \Vest. A. S..Ibid.. 42, 2303 (1950). (8) Dunlap. I. K.. Rushton. J . H.. Spmp. Ser. No. 5. 49, 137 (1949). (9) Eisenberg. M., Chang. P.. Tobias, C. \V., LVilke. C. R . ? il.1.Ch.E. J . 1 , 533 (1955). (10) Lisenbmq. M.. Tobias. C . \I-..IVilke. C . R.. "Mass Transfer at Rotatinq Electrodes." Part 11. Department of Chemistry a n d Chemical Cngineering, University of California, September 1353. (11) Harriott. Peter. Ibid.. 8, 9 3 (1962). (12) Hixson. .A. \V.. Bauni. S.J.. Ind. Eng. Chem. 33, 478 (1941). 113) I h d . . 34. 120 11942). (14j Hixson. '.A. \V.', \\-ilkens. G. A., Ibid., 25, 1196 (1933). (15) Johnson. A. I.. Huang. C.. A.I.Ch.E. J.2 , 412 (1956). (16) King, C . V., Brodie, S. S., J.Am. ChPm. Soc. 59, 1375 (1937'1. (17) Levenspiel. O., \Veinstein. N. J., Li. J. C. R., I n d . Eng: Chem. 48, 324 (1956). (18) hfarangozis. J., Johnson. A . I., Can. J . Chem. Eng. 39, 152
lb cu ft
Pro 3o Sh/Sco '0
= Su
literature Cited
(1) Askew, L+-. S.. Ph.D. thesis. University of Maryland. College Park. Md.. June 1965. (2) Barker, J. J.. Treyhal. R. E., "Mass Transfer Coefficients for Solids Suspended in Agitated Liquids,'' 41st National A.1,Ch.E. Meeting. St. Paul. hiinn.. 1959. (3) Bates, R. L.. Fondy. P. L.. Corpstein, K. R.. I r a . ENG.&EM. PROCESS DF.SIGN DEVELOP. 2 , 310 (1963).
(iij'iiih.. 40, 231 (1962). (20) Oldshue. J . Y.,Gretton, A. T.: Chem. Eng. Progr. 50, 615 (1 954). (21) Rushton, J. H.. Costich. E. \V.? Everett, H. J.? Ibid.. 46, 395. 467 f1950). (22) -Rushtdn. J.'H., Lichtmann, R. S.. Mahony, L. H., Ind. Eng. Chem. 40, 1082 (1948). (23) Seider. E. iY., Tate, G. E.. Ibid.. 28, 1429 (1936). (24) Sherwood. T. K.: Ryan. J . M., Chem. En,q. Sci. 11, 81 (1959). (25) Chi. V. \V.. Heat Transfer Symposium, Annual Meeting A.1.Ch.E.. 1953. (26) Volk. \$Xiam. ".Applied Statistics for Engineers," pp. 266269. McGraw-Hill. New York. 1958. RECEIVED for re\iew August 18, 1964 RESUBMITTED October 22, 1964 ACCEPTED March 12. 1965
BLENDER GEOMETRY IN T H E MIXING
OF SOLIDS D. S. CAHN, T. W .
H E A L Y , AND D. W . FUERSTENAU
Dfflartment of 'Mineral Technology, Criiterstty of California, Berkeley, Calzf,
A variable-V mixer i s described and used to isolate the mechanism of mixing of particulate solids in V mixers or twin-shel! blenders. From a consideration of the geometry of the system, a geometrical mixing p a rameter in V mixers i s proposed and correlated with a conventional statistical criterion of mixedness. The ina standard V mixer-to creased rate of mixing observed as the variable angle increased from zero-i.e., 1 80°-i.e., an inclined barrel mixer-is interpreted in terms of'flow between the arms of the mixer. The effect of loading arrangement i s examined and correlated with the geometrical properties of the system. the mixing of solids has been \\idely used in chemical, ceramic. and metallurgical processing for many years, it is only recently that significant efforts have been undertaken to understand the detailed mechanisms of the mixing process. I n general, the research reported in the literature has been oriented either toward machine technology or to\vard theoretical analysis, often w.ith inadequate experimentation on well characterized systems to test the theories. T h e approach in this present study has been mechanistic rather than mathematical. T h e mathematical theories of the rate of mixing of solids. a t their present stage of development, are applicable only to limited regions of the rate of mixing LTHOUGH
A
318
I&EC PROCESS D E S I G N A N D DEVELOPMENT
curves for ideal systems. Since real systems differ from this ideal case, it is important to isolate and describe the principal mechanisms of mixing and to assess the contribution of each mechanism in any given mixing operation. T h e \' mixer, or tu.in-shell blender, in which the bed of particles is split and redeposited during each cycle leads to rapid mixing. This mixing is referred to as convective because it involves movement of groups of particles from one position to another. I n contrast to this, diffusional mixing takes place as a random movement of individual solid particles past one another in the shear-gradient zone created in the mixture by the tumbling action of the blender. A horizontal barrel
VARIABLE V MIXER ANGLE ANGLE
P
+
FIXED
P VARIABLE
OF M I X E R
0.
L'
4-P.0"
CONDITIONS V-MIXER, Q = O " SIZE-SMA1.L 715cc
0
24 rpm 0 0
0
20
40 VOLUME CHARGE,
L U C ' T E BALLS GI b S S RAILLS
PO
60
100
PEiiCENi FILLING
Figure 1 . Schematic representation of variable-V mixer showing various settirigs of variable angle /3
Figure 2. Kinetics of mixing of Lucite and glass balls as a function of volume Filling of a standard V mixer
mixer loaded end to end-- Le.. the two components initially separatrd by a plane normal I O the axis of rotation-- mixes bl- a diffusion mechanism. 'I he rrirchanism of mixing in V mixers has been ctudied by several Lvorkers (7--3: 6 ) . LViedenbaum, Corson, and Miller (6) and Kaufinan 1.3) have surveyed the statistical aspccts of saiiipling in a c' mixer and V mixers and other tumbling type blender,!;have been compared. hlixing zones withir: the \' mixer observed by IVeidenbaum ~t ai. ( 6 ) show clearly that a convective mixing mechailism is operating. ' l o analyze this mechanism in detail, a variable\.: mixer has been constructed in which the t\vo arms of the mixer can be adjusted so that each arin is in a plane rotated a variable number of degrees from the other. The V mixer, \iith the arms rotated 180'. becomes a n inclined barrel mixer. ' l h e prime reason for constructing the variable-V mixer \vas to provide a means for isolating varioiis rnechanisms of mixing in tumbling mixers in general. LVitli this innovation, it has been possible to isolate the ineclianism of mixing in the V mixer and to express the nature of convective mixing quantitatively. I t h a s also been possible to assess the statistical criterion of mixing based on the r.tandard deviation among spot samples in terms of geometrical parameters that deccrihe a mixer. its contents. and the motion kvithin the mixer.
mum rate o r efficiency of mixing occurs in this V mixer when the mixer has been filled t o about 40$7 of its total volume. and this loading \\as then ust-d for subsrqiient esp:.' 6 I lnlrIltS 011 mixing partictilatr materials. I n the study of the mixing of particulate Inaterial. 2 X - X 35mrsh C:aCOa and Si02 Liere loadt-d Side by side o r as laycrs at the base of the 1-of a 1078-cc. variable-\- mixer. Variable angler for the sidr b>-sidr loading wrie 0'. 20°, 40@,6 0 ° , 8O@, 1 2 0 @ ,and 180'. arid 0' and 60' for thr laycr-loaded mixer. 'I'hr angles of repose of Sic).,. Cla(:O:l. arid a i aridoin mix of SiO, and CaCOr were 34". 3 0 @ ,a i d 3 2 @ ,respectively. For the loading of 40% capacity, 291 grams of (bulk density 1.35 grams per cc.) and 261 grams of Si02 ( h i i l k density eqiial 1.21 grams per cc.) \\ere used to gike appr(~~iir~arc.ly nurnberr of par ticles for each component. Samples of the mixture were t;tken ivith a slwve-type sampling rod of I , ?-inch outer diameter. and appi-r)xirriately 0.25 gram maximum cciuld be removed for each cample. T'he mixer \vas loaded \vith the appropriate Lsrights of calcite and quartz, Lvith the calcite in the right c' and quartz in the left 1' in the case of axial loading. A partitioil iriserred to aid in the filling of the mixer \vas ren1ovc.d and the iriixcr \vaq rotated for the appropriate number of revolutions. A t each iiuinber of revolutions (mixing time) and at each setting of the variable angle, the mixer \vas stopped and the eight samples ivere Ivithdrawm. (Details of the statistical aspectc of sample size: number of samples. and the expressioii of mixrr analysis in terms of statistical criterion of mixing are give11 belo\v.) Each sample was then Lveighed and from i to 1 0 nil. of 0.5,M I {(:I \vas added. 7'he sample plus hbdrochlorir acid solutiori \\-as heated over a low flamt- for 7 to 8 minutes and thrn hack-titrated Lvith standard S a O H t o a phenolphthalrin e n d imixit. O n the basis of the volume of kno\vn molarity acid used to diswlve the calcium carbonate. the \\eight and pt-I crnt calcite in each sample were deterinined. 'l'he sainple variance of each set of' eight samples was the? calcu!ated in r v i . I I i h of the n i i m l ~ e rof C:aC08 particle4 per sanil~le.
Experimental Method 'and Materials
A schematic drawing of the variable-V mixer with the arms set a t different position:: defined hy the variable angle 13is given in Figure 1 . Preliminary studies of the efficiency of mixing kvere carried out \sit11 .',-inch-dianieter Lucite (specific gravity 1) or glass balls (specific gravity 2.6): and the kiiietics of mixing \vere examined by determining the average number of revolutions for a marked ball to move from one end of the mixer to the center. 'l'lie movement of the marked ball was determined as a function of the mixer loading and of the angle betiveen the arnis of the mixer. l'liese qualitative studies weie carried out with the variable angle.. P! set at O", 20'. and 40@; the V mixer (0.7-liter capacity) used in this aspect of the study was operated a t 24 r.p.m. for various loadings of the mixer \vir11 glass or Lucite balls. Data for the average number of rotations required for one marked bead in a batch of marked beads to move from one end of the mixer to the center as a function of mixer filling \Yere normalized to 50TC volume for each mixer filling and replotted against mixer filling. A maximuni rate of movement (minimum r1umber of 1.t.vo1utions for the bead to traverse the disSho\vn in Figui,e 2. C'learly, thr maxitance) i r obseivcd,
Mixing of Particulate Matter in Variable-,V Mixer
'I'he preliminai y stud) oiitliried above coilCirInr3d the coiicept of a variable-\- mixer. I t also sho\\ed that a m i x ~ iloading o f 40%, filling gave optirnuiii mixing and rh:rt change in the variable angle effected more rapid niiwing. \\.itti this barkground. it \vas possible to design a st-t of e x ~ ~ ~ r i i r i eini i\vl~ich t~ particulate materials \\ere mixed. Statistical Aspects of Mixing. 'The ohserved degrrc of mixedness at any angle 8 is charactei~izrd b) a parameter: iii the folio\\ irig ielatioiiship iI 6):
VOI
4
140
3
IULY
1965
319
SIDE - B Y - S I DE LOADING V A R I A B L E ANGLE p
t
LAYER LOADING 0 O', 6 0 ° = 4
1
\ \ 1 10
I
100
1000
NUMBER OF R E V O L U T I O N S , N
Figure 3.
NUMBER OF REVOL
Progress of mixing of C a C 0 3 and S i 0 2 particles
Expressed os standard deviation, crx, v s . number of revolutions, N, of variable-V mixer for side-by-side (axial) loading and for various settings of voriable angle
Figure 4. Mixing of C a C 0 3 and Si02 particles loaded side by side and layer-loaded
where calculated equilibrium variance of a random mixture calculated variance of unmixed system uNz = sample variance of the system for any number of revolutions, 'V. of the mixer
probability of
M N then
is equal to (1 - a ) , where
uT2 = uo2 =
goes from 0 to 1 as the variance of the system goes from U: (unmixed) to ur2 (mixed). T h e above variances are defined by = PA(I
- Pa)
(2)
wherep, = number fraction of particles of type A in the system of A and B particles.
where n = number of particles in each sample. An unbiased estimate of the true variance of the system at any number of revolutions, N , is given by S
u.v
-
2=1
-
s - 1
(4)
where x I = fraction of particles of type A in the sample
R = average fraction of particles of type A in all samples taken S = number of samples taken Accordingly, the criterion of mixedness, M v , is directly the sample variance ( I n the definition of M v , related to uV2% theoretical variances are implied. but it must be remembered that u\.~, u , ~ ,and uo2 are "observed" or calculated estimates of the corre$ponding theoretical variances) In terms of the number fraction of C a C 0 3 particles in the system, the theoretical (unmixed and random variances) are go2 =
PC.CO,(~- PracoJ
(5)
T h e sample distribution was assumed to be normal with mean (PA) and unknown variance, uz (of lvhich is the unbiased estimate), for each time-angle combination. T h e (1 - a ) confidence region for u2 is then defined such that the 320
I&EC PROCESS DESIGN AND DEVELOPMENT
(7)
S
= number of samples to be taken for each time-angle
uN2
= sample variance after
combination
x2
~ ( ~ ~ -=1 ) ~random
S revolutions of the mixer variable on ( S - 1) degrees of free-
dom For a high per cent confidence-Le., small a-the number of samples, S, required to satisfy the above relationship is large. I t is, therefore, necessary to balance the size of the confidence interval against the time of taking and analyzing a large number of samples. In the present study, the optimum a LIS. S combination arrived at in this \yay was S = 8 and a = 0.20. For this combination, there is a n 80% probability that the true standard deviation, u: lies between 0.74 and 1.58 b y . Weidenbaum (6) recommends taking a minimum of 14 samples for each point. However, the number of particles per sample (which directly affects the accuracy of the sampling method) in his case was one fifth that of the present work. We have shown that eight samples are suitable for our conditions. Because of the irregular shape of the mixer interior at large angles of rotation, the volume of the batch that could be sampled was limited by the diameter and filling length of the rod sampler. Twenty-six possible sampling positions, 1 3 on each V, were selected. From these, four positions were selected a t random for each side. As angle /3 exceeded 90°, it was difficult to maintain the same sample pattern. Based on an average sample weight of 0.2 gram, the number of particles in a randomly mixed sample was found to be approximately 850. Results. The kinetics of mixing of CaC03-SiOz, expressed us. the logarithm of the number as the standard deviation, uAT5 of revolutions, S, are shown in Figure 3> for settings of che variable angle p of O o , 20°, 40', 60°, 80', 120', and 180'. T h e curve for 180' is dashed, since the mixer cannot be sampled at this angle in the same way as at other angles. T h e effect of layer loading the V mixer a t values of /3 of 0 ' and 60' is shown in Figure 4. An end-loaded V mixer at a setting of the variable angle of = 120' behaves in the same
Figure 6. OOOlL__i__i_ 0 20
40
I
,
60
80
VURIABLE ANGLE,
Figure
,
I
1 ,
100
120
140
I
1
160
180
b , DEGREES
Geometry of variable-V mixer
Expressed in terms of v a r i a b l e a n g l e @, a n g l e of r o t a t i o n of the mixer,w, and 8, a n g l e defining Row
5. Effect of variable angle p on statistical criterion
of mixing M., Expressed as ( 1
-
M., i for varlous numbers of revolutions of mixer
\vay a s a layer-loaded 'v' mixer at
/3
=
0 . Visual examination
cif thr mixer itself sho\ir that at p = 120' the end-loaded mixer transforms to layer-loading after one revolution. \'ariation in angle /3 does not affect the rate of mixing in layer-
loaded systems. The effect of varying angle /3 on the kinetics of mixing is sho\vn in Figure 5, where the logarithm of the statistical criterion of mixedness, hl,, expressed as (1 is plotted as a function of the variable angle, B: at various times of mixing. Discussion
Statistical Analysis of Data. T h e linear regions of the S plots of Figure 3 can be represented as
u., us. log
u . ~ = a
log .V $- h
(8)
where a is the slope and b is the intercept (at a value of log ,V = 0 or .V = 1 revolution) of the linear regions of Figure 3. l a b l e I sumrnarizes the least squares values of a and b a t various settings of the variable angle. T h e data for p = l;!Oo are separated in Table I , since it is shown below that for /3 > 90' there is a change in the mechanism of mixing. If the intercept, b , at log -V = 0 is plotted against the variable angle, 8, a straight line is obtained for values of @ of O', 40°, 60'. and 80'. Again, the point for 120' does not fall on this straight line. T h e complete empirical relationship among u . ~.V, ) and B is therefore given by n.v = -11.77 log
S - 0.212 fl
+ 74.11
(9)
for 0 5 /3 5 90' and for the range cf number of revolutions given in the last column of Table I. (The coefficients of log and 0 and the constant of Equation 8 were obtained by standard statistical analysis of the data of Table I for variable angles of O', 40°, 60', and B O O . ) Similarly, the log (1 - M,v) us. fl plots at various number of revolutions, S,shown in Figure 5 can be expressed by a n empirical equation of the form of Equation 8. T h e three empirical constants in Equation 9 clearly depend on variables not examined in this present investigation-i.e., mixer speed. dimensions, and loading. Geometrical Mixing Parameter. Consider the motion of rnixer during each cycle. At fl = 0 for the charge within a \7
Table I . Empirical Values of Parameters of Equation 8 Vat~abl~ Range of Angle, Intercejt, d.b,bllCa~IO71. Iltgrres, 4 Slop 0i
riglit is
dit,
$1,
i 0
(loa)
and the conditiun for flo\v light to left is $13
> 0 > $L,
$R
>8
(lob)
where
Acknowledgment
?'he authors thank M. J. Yokoca. C. R.Guerra, and other graduate students in the Department of Mineral Technology for assistance in development of this project. Experimental assistance from John Erickson and M'illiam Lee is acknowledged. Nomenclature
and w =
6
=
0
=
angle of rotation of the mixer variable angle 13.5" for the present s)-stem of CaCOa--SiOn at 40% filling, as measured experimentally
From plot> of Equations 1 l a and 11b and application of the conditions given in Eqiiations 10a and lob, the periods of floxv during each cycle 'of the mixer for each setting of the variable angle 6 can readily be calculated. From the resulting f l o ~ v diagram, a grumetrical mixing parameter can no\\ be defined for the variable-V mixer. Let Q be the fraction of each cycle during ivhich flow occurs, either left to right or right to left. Then Q variesfrom 0.005 at /5' = 20' to 0.80 a t /3 = 180'. T h e geometrical mixing parameter, Q, developed above drfinrs the fraction of earh revolution of the mixer during which flow takes place betxveen the arms both left to right and right to left. 'I'hat this criterion is proportional to 'if.\-,the criterion based on statistical arguments, is shown in Figure 7 , xvhere Af.v (experimental) for three rimes of mixing and Q (theoretical) are givcn as functions of the variablr angle. p. T h e family of curves shown in Figiirr 7 indicate that M,. and Q are dirrrtl!' proportional to earh other as AI',
=
Q
-+ constant
(12)
I'he rarigcs of application of this rquation are once again the rangrs given in l'able I . Visual examination of t h e V iriixei, set xvith the variable angle $ at 120" shoivs that at this angle the end-loaded mixer trans322
forms after one revolution to layrr loading. T h e data of Figure 4 confirm this and also sho\v that with layer loading there is little change in the mixing as the variable angle is Slixing in the l7 mixer under incrcased from 0" to 60'. layer-loading conditions probabl), occurs principally by a sl'litting-reforming mechanism and to a lesser extent by a diffusional mechanism. 'I'he slope of the u . ~21s. .Y curve for layer loading (Figure 4) is about the same as the slope obtained for side-loaded condirions. Because of the complex mechanisms f(Jr mixing la>.er-loadcd systems. no qiiantirative explanation is yet available. .An important concliision that arises from the correlation can be stated cither as (1) the validity of the geometrical mixing parameter. Q,based on mutual flo~vbetween the arms during each cycle: is verified by the fact that Q has been shown to be proportional to the s~atistical criterion of mixedness, Aft-; or (2) the statistical criterion mixedness, ,i4.\-! is directly proportional to Q . a parameter that defines the interflo\v (mixing) during each rotation of the V mixer. Both are arbitrary parameters by definition, so that the importance of the correlation is that Ms and Q are tsvo independently derived but consistent parameters.
I&EC
P R O C E S S DESIGN AND DEVELOPMENT
statistical Criterion of mixing number of revolutions of mixer number of particles in each sample geometrical (flow) criterion of mixing number fraction of particles of type A in a system of .4 and B particles number fraction of C a C 0 3 particles in C a C 0 3 S O 2system number of samples fraction of particles of type i average fraction of particles in all samples taken per cent confidence level variable angle of lrmixer angle a t which flow occurs for a given geometry of mixer and its contents standard deviation among S samples at a number of revolutions 'V variance of same calculated equilibrium variance of a random mixture calculated variance of unmixed system chi square random variable on ( S - 1) degrees of freedom gradients of left a r m and right a r m axis of the mixer angle of rotation of whole mixer literature Cited
(1) Carley-Macauly, K . LV., Donald, M. B., Cheni. Eng. Scz. 17, 493 (1962). (2) Gray. J. B., Chent. EnE. Progr. 53, 25 (1957). ( 3 ) Kauftnan, .4.. IND.ENG.CHEM.FUNDAMENTALS 1, 104 (1962). (4) Lacey, P. M. C., J. -Ippl. Chem. (London) 4, 257 (1954). (5) iVeitienbauni, S. S . , Adr~an.Chon. Ens. 2, 211 (1958). (6) \Veidenhaum. S. S.. Corson. R. C.. Millcr, D. P., Crrainzc Aqe 79, 39 (1963). RECEIVED for review October 19, 1964 ACCEPTED April 12. 1965 IVork supported by the United States .4ir Force. Aeronautical Systmis Division, under Contract No. A F 33(616)-7763.
