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Longitudinal mixing of granular material flowing through a rotating cylinder
Chemical Engineering Science, 1965, Vol. 20, pp. 1089-1100.
Pergamon Press Ltd., Oxford.
Printed in Great Britain.
Longitudinal mixing of granular material flowing through a rotating cylinder Part II. Experimental R. RUTGERS Laboratory
Royal Lassie Mills Ltd., Wormerveer,
the Netherlands
(Received 3 February 1965) Abstract-Experiments on axial dispersion for continuous flow of, e.g. long grain rice through a laboratory-scale rotating cylinder are reported. The model of piston flow with superimposed longitudinal diffusivity proved to be valid. Axial dispersion coefficients were determined under varying conditions of loading, speed of rotation and number of rotations, linear flow velocity, inclination, types of particle movement, shapes of cylinder entrance and exit end faces. Furthermore, the effect of some variations in the properties of the solid particles has been studied e.g. particle shape, particle size distribution and particle cohesion. Some experiments with a cylinder of larger diameter are reported. Longitudinal dispersion coefficients could be rather low, e.g. down to 0.04 cm2/sec, much lower than most other types of fluid flow processes and reactors.
1. EXPERIMENTS EXPERIMENTS are described for a small laboratory
scale cylinder of perspex with L = 50 cm and D, = 16.0 cm (NC = 106 r.p.m.), with varying dimensions and shapes of the inlet and outlet sections, involving variation in the working content (see Fig. 1). The straight discharge end faces were made of aluminium and all other end faces of Perspex, sometimes covered with emery paper. For most tests the inner wall of the cylinder was covered with emery paper, to prevent slip. In some tests the smooth wall was used and surging took place. The speed of rotation was varied between 8 and 97 r.p.m. or O-133 and l-62 r.p.s. In one test the drum was given an inclination of 4”; for the other experiments the cylinder had a horizontal position. With some tests 4 smaller wooden radial ribs, extending over the entire length, of 0.6 cm thickness and 05 or
FIG. 1. Experimental
1.2 cm height were inserted, equally spaced over the circumference. In some tests the effective length of the cylinder was halved. As granular material, mostly air-dry free-flowing long-grain milled rice with a moisture content of 13 % was used, with particle dimensions of O-70 x 0.21 x 0.17 cm. The average kernel weight was O-0188 g with a standard deviation of only 0.0017 g. Some tests were performed with sticky rice and with a mixture of whole and broken kernels. As another granular solid the more angular cut oat groat particles were used. Some of the grains were coloured red with fuchsine, care being taken not to change their surface properties. All tests were done with continuous flow and with vibrator feeding. The feeding rate could vary somewhat during a test, e.g. +5 ‘A. After reaching a steady state with white rice, red rice was fed, for practical reasons for about O-4 of the average
drum with inlet and discharge variations.
1089
R. RUTGERS residence time, followed again by white rice. The residence time used was mostly about 600 set, with some variations. The working content was checked repeatedly. At the outlet samples of about 5-20 g were taken for about 5-10 set, with a time interval of 15-30 sec. This method of sampling would seem to approach point sampling near enough to render it unnecessary to introduce an extra variance. For a number of experiments the whole response curve was determined, whereas for the other tests only the rising branch of the exit concentration curve up to a concentration of 0.5 was determined. The content of red grain was determined by visual selection by hand and expressed as a weight fraction, which in this case is identical with a volume or number fraction. Recovery of red grains was satisfactory. No serious contamination of the white kernels took place.
2. CALCULATIONS The exit concentrations of red-coloured grains were put in t-F response curves at the average time of the duration of the sampling. Smooth lines could be drawn through the points. As for most tests red kernels began to appear after about +? and red rice was fed for O-42, the exit concentration should be usable up to about 0*85i (up to F-values of e.g. 0.35) for the step-function calculation of the Bodenstein numbers. The exit concentration tables of BRENNER [I], based on finite vessel length, were used for deriving Bofigures from the experiments. Taking 3 or 4 points of the ascending branch of the response curve gave Bo-figures which were reasonably constant in most cases. This confirms the validity of the chosen diffusivity model for the present type of flow of granular material. The average spread from the mean Bo-value was + 5 %. There were some exceptions to the constancy of the Brenner Bo-figures, which will be discussed later on (Tests 7, 9, 15, 24 and 25). For a number of experiments Bo values were calculated with the Danckwerts formula (See Part I). The results were 20-15 % lower, in the range of Bo from 20-100, than the Brenner results, and there was a definite trend towards lower Danckwerts Bo’s when calculations were made for longer times in a
test. This model is not the most appropriate for our case, even for higher Bo numbers. The total exit concentration curves has the familiar bell shape with a tendency to tailing. Two examples, Nos. 13 and 16 are given in Fig. 2. The different shape for Tests 7 and 9, which indicated a different flow pattern of the grains, will be discussed later on. The cumulative concentration curve on a probablility scale against time on a logarithmical scale has been shown for some examples in Fig 3. Up to 90-95x rather straight lines are obtained, indicating a log-normal distribution, especially for higher Bo numbers. Applying the criterion of VAN DEEMTERet al. [2]. for a feeding time of marked material which is possible without changing the width of the response curve, we find O*lf for a Bo number of 50 (using mixing cell length as the theoretical plate length). It is seen that a feeding time for coloured grain of 0.47 is too long to allow the use of pulse function formulas for the calculation of Bo numbers. KLINKENBERG [3] suggested statistical treatment be applied for the calculation of the Bo number in the case of slug feeding, using the whole response curve. The distribution curve is transposed into the cumulative curve, using $-min intervals. By numerical partial integration according to SIMPSON with I-min intervals, the average residence time T including half of the feeding time t, of coloured grain is found:
7 = -
G
Qlll
is the average residence
time
of grains
in
the cylinder. By numerical integration of t( 1 - F) the variance of the concentration-time curve is found from : (1 - F)t dt - T’)
(2)
S2 is the sum of the relative variance a2 of the residence time of the grains in the rotating cylinder and the variance of the feeding time of coloured particles,
1090
which is z: s2 = 02 +
2
(3)
Longitudinal
mixing of granular material flowing through a rotating cylinder-part
t FIG. 2.
-$- 1 -
$0” - $(02)2 - gr’)”
see
Typical response curves.
From c2 the Bodenstein number can be calculated according to Formula (2) in Part I, which is made easier for calculations by Klinkenberg in the following transformation: Bo =
II
(4)
This calculation was carried out for a dozen experiments. The calculated feeding times of coloured grains was on average a fraction of 0.83 of the actual times used. The Bo values were 0.89 times the numbers calculated from the ascending branch of the distribution response curve with the use of the Brenner tables. With regard to the validity of the models used as a basis for calculation of 02, of Bo from Formula (4) and of the dispersion coefficient M1 it should be borne in mind that some irregularities are apt to occur at the entrance and at the exit of the cylinders. With vibrator feeding the change from normal grain to coloured kernels and back to white again involves some variations in feeding velocity. At the entry coloured grains tend to be fed over a certain length of the cylinder and not at a point,
and furthermore it takes some time to distribute them evenly over the cross-section by radial mixing. At the outlet the grains strike against the vertical end-face, which pushes them back, causing swirls and eddies which increase local axial mixing. The tail of the response curve will be made longer. Furthermore the grains must leave the cylinder through a narrow opening under the pressure of a considerable head, whilst the particles are moving radially. The effects of various forms of the endfaces, as shown in Fig. 1, have been investigated and will be discussed later on. Most experiments were done by measuring only the first part of the response curve, and the Bo numbers used for calculation of the dispersion coefficients in cm2/sec, according to Formula (5), were derived from the Brenner tables.
3. &SXJLTS In Table 1 the cumulative response curve figures are given for a number of experiments. Table 2
1091
R. RUTGERS
360 240
-13 j-5
63 13.4 15 /0 0
x-__---_,6
120
o__--_- 7 ~________ cJ 0001
0.010
0.100 0.200
0.500
0.8000.900
0.990
0,999
F FIG. 3.
Table 1. 6
0400
oaoo
O-000
0.000
0900
OMO
0400
0400
O-000
0400
0.000
0400 0400 0400
0.002 0.009 0.026
0900 0400
0400 0900
6 7 8 9 10
0.001 0.023 0.088 0,195 0.348
0.060 0.121 0.217 0.345 0.471
0.000 0300 0.006 0.046 0.158
11 12 13 14 15
0.535 0.712 0.839 0.915 0.956
0.583 0.677 0.753 0.814 0.862
16 17 18 19 20
0.979 0.991 0.996 0.999 -
21 22 23 24 25
-
Time (min) 0
2
Cumulative response concentrations
5
Test No.
o*ooo
Cumulative response curve.
11
12
13
14
0400
0.000
omo
0.000
0.000
0400
0400
0.000
o*ooo
0400
O-000
OGOO
oaoo
o*ooo 0900 omo o*ooo
0000
0400
0.000 0000 0.000
0400 0400 0.000 0400
OGOl OW8 0.038 0.107 0.213
0.002 0.016 0.056 0.139 0.265
0*004 0.024 0.082 0.192 0.354
0.001 0.011 0.045 0,124 0.249
0.352 0.583 0.763 0.875 0.931
0.345 0,496 0.637 0.756 0.847
0.431 0.602 0.747 0.848 0.916
0.549 0.729 0.853 0.917 0.954
0.897 0.923 0.945 0.961 0.972
0.959 0.977 0.987 0.992 0.995
0.911 0.951 0.974 0.986 0.994
0.955 0.977 0.987 0.993 0.997
0.975 0.987 0.994 0.997 0,999
0.979 0.985 0.990 0.993 0.996
0.997 -
0.997 0.998 -
0.998 0.999 -
o*ooo
8
omo
10
o*ooo omo oaoo
-
1092
16
17
0400
0.000
OGOO
0400
0900
0400 0400
0400 0300
0400 O+lOl
0400 OGOO 0400 0300 0.003
0.003 0.023 0.089 0.226 0.420
0400 0.003 0.024 0.081 0.191
0.005 0.033 0.112 0.255 0447
0.017 0.053 0.120 0.213 0.327
0.002 oGO9 0,035 0.087 0.165
0.419 0.592 0.732 0.835 0.903
0.634 0.804 0.904 0.954 0.974
0.349 0.542 0.719 0.851 0.926
0.627 0.762 0.854 0.912 0.949
0.452 0.571 0.675 0.763 0.832
0.270 0405 0.552 0.690 0.800
0.946 0.971 0.984 0.990 0.994
0.986 0.994 0.997 0.999 -
0.965 @984 0.993 0.997 0.999
0.972 0.985 0.992 0.996 0.998
0.883 0.921 0.947 0.963 0.974
0.877 0.922 0.951 0.970 0.982
0.997 0.998 -
-
-
-
0.980 0.985 0.989 0.992 0.995
0.988 0.992 0.995 0.996 0.997
-
1.5
oaoo 0400 0400
Longitudinal
mixing of granular material flowing through a rotating cylinder-Part
gives details of these tests together with those of other experiments with the same type of long dry whole rice kernels; Bo and MI figures are mentioned. For the indications of various entry and discharge openings, see Fig. 1. Furthermore the speed of rotation in r.p.s., the working content G in grams, the feed rate Q, in g per set, the average residence time i in sets and the time of feeding coloured rice 0, as a fraction of 7 at the same flow rate are recorded. The dispersion coefficients are calculated
Table 2.
II
for experiments with conical end-faces with L = 50 cm (see below). The fractional volumetric degree of filling P of the cylinder was approx. O-15 for a content G of 1000 g. This factor was somewhat lower for higher loading and also lower when surging occurs (Experiments 7 and 9). Higher speed of rotation, causing lower dynamic bulk density, increases P (Experiments 8, 16). Further experiments with grains of varying particle properties will be discussed later on.
Experimental results -
No.
Inter
Outlet
14 13 32 12 2 57
r.p.s.
Variations
G(g)
Qm 63/w)
C (set)
0,
Bo
MI
(cm2/sec)
0.50 0.50 0.50 o-50 0.50 0.50
-
4010 2620 2750 1390 1760 1575
6.55 4.85 4.32 2.32 3.15 2.37
612 540 637 600 559 591
0.38 0.41 040 0.39 0.38 0.39
:: 47 46
0.064 0.074 0.083 0.091 0.095 0.092
63 63
6 8 58 16
a a a a
3 3 3 3
0.133 1.00 l*OO 1.62
-
1620 1635 1850 2490
2.50 2.37 3.17 4.03
648 613 584 612
0.37 040 0.41 0.39
89 44 33 15.0
0.044 0.093 0.129 0.270
15 :1”
a a a
1 3 3
0.50 0.50 0.50
CL< = 4” ribs, 0.4 cm ribs, 1.2 cm
2030 1340 1305
3.60 2.23 2.28
564 600 572
0.40 0.41 040
35 38 42
0.127 0.110 0.104
4 5 7 9
a a a a
0.50 0.50 0.50 0.133
L=25cm L=25cm smooth wall smooth wall
770 730 1510 3300
2.30 1.37 2.47 2.38
335 548 607 1385
0.85 0.31 040 0.27
20 13.4 ? ?
0.093 0.085 ? ?
22 :: 47 2 61 62 23 46 30 ;: 24
b, sm b, sm b, sm b, r b, sm b, sm b, sm b, sm b, b, b, b, b, b,
sm sm sm sm sm sm
4, sm 4, r 5, sm 5, r 6, sm 6, sm
0.50 0.50 o-50 0.50 0.50 0.50 0.50 0.50
-
2595 2580 2520 2550 1600 1440 1325 1300
4.33 4.50 4.08 4.17 5.47 2.25 1.07 0.52
599 573 617 612 294 640 1284 2516
0.39 040 040 040 0.80 0.41 0.29 0.26
105 98 76 58 120 52 34 20
0.041 0.045 0.053 0.070 0.071 0.075 0.058 0.050
050 0.50 0.50 0.50 0.50 0.50
-
2510 2640 2150 2270 1425 1385
435 4.35 3.98 3.78 2.27 2.22
577 606 540 600 629 625
040 040 0.40 0.41 0.39 0.55
55 65 33 59 18 23
0.079 0.063 0.140 0.071 0.220 0.174
sm = smooth (Perspex.) Ml figures from No. 22 on are calculated with L = 50 cm (see text). r = rough (emery paper.)
1093
R. RUTGERS
4. 4.1
the dispersion coefficient to I/ JF or l/ JG.
DISCUSSION OF THE RESULTS
Reproducibility
Table 3 contains some data for M1 on the repetition of experiments, which give an idea of the accuracy of the determination and the calculation of the dispersion coefficient in the way which has been described. Table 3.
Reproducibility Ml
A
13 32
0.074 0.083 >
0.009
2 57 12
0.095 0.091I0,091
0.004
8 58
0.093 0.129 >
0.036
45 48 22
0045 0.053> 0.041
0.012
25 24
0.220 0174 >
0.046
2, 12, 57 5 59 60 61 62
2,12,57 13,32 14
a
Outlet b, smooth
b” :: b
4.4
1575 583 730 548 1600 294 1440 640 1325 1284 1300 2516
0.086 0.046 0.170 0.078 0.039 O-020
0.093 0.085 0.071 0.075 0.058 0.050
Speed of rotation and number of revolutions
The results for experiments with an increasing speed of rotation at constant residence time and the same inlet a and outlet 3 are collected in Table 6. Here the ratio of N to the critical speed of rotation NC, the ratio M,/&?(N in r.p.s.)‘and the Froude W2R
Influence of loading
0.25 0.41 0.58
u MI (cm/set) (cm+ec)
values for a greater length and also probably for a longer residence time may result from the relatively smaller influence of the end effects on the overall Bo and MI figures and is indicated in the M1 figures found. However, the somewhat lower working content at a lower feed rate will have had the opposite effect on M1.
number -
1575 2685 4010
50 25 50 50 50 50
t (set)
For a range of u between 0.02 and 0.17 cm/set there is only some variation in the dispersion coefficient. The tendency to obtain lower M1
mentioned.
P
L (cm) G(g)
N = 60 r.p.m
The degree of volumetric filling P has been calculated from the working content G and the dynamic bulk density, which varies slightly with the working content. The results for tests with 30 r.p.m. are given in Table 4, where the product M,*,/P is also
G(g)
Influence of axial linear velocity Inlet
Test
Remarks
Loading G, degree ofJilling P
Test
Axial linear velocity
Table 5.
Somewhat greater variation is observed at a speed of rotation of 60 r.p.m. in the transition region prior to cataracting and with the smooth longish conical outlet 6, which gave more gliding, backswirling and local axial mixing.
Table 4.
proportional
Table 5 gives the results for some tests with variation of length and residence time, all for N = 30 r.p.m. and outlet No. 3.
of experiments
Test No.
4.2
4.3
is approx.
Ml
are also given.
Table 6.
Ml4
(cm2/sec)
0.093 0.078 0.064
9
No.
0.047 0.050 0.049
It is seen that for the chosen moderate speed of rotation with the rolling type of particle motion,
6 2,12,57 8,58 16
1094
N
Influence of speed of rotation N
N
+= G
MI
(r.p.s.) (r.p.m.) NC (s)
(g)
0.133 0.50 1.00 1.62
1620 1575 1740 2490
8 30 60 97
0.075 0.28 0.57 0.92
648 583 598 612
MI
Froude
(cmQec) ~‘8 number 0.044 0.093 O-111 0.270
0.12 0.13 O-11 0.21
OT1057 0.080 0.32 0.84
Longitudinal
mixing of granular material flowing through a rotating cylinder-Part
The working content was rather constant up to 60 r.p.m. and only minor corrections of Mr for varying content would be needed. Under cataracting conditions G was considerably higher. For a lower content M1 would be higher still, but it has not been investigated whether a proportionality with l/,/G holds under these conditions too. Up to about 60 r.p.m., the speed of rotation region where cataracting started for this type of grain, M1 increases in proportion to approx J??. At a higher speed a much sharper increase of the dispersion coefficient is found. Close to N, a decrease of M1 might occur. Further investigations into the relation between M1 and N in the higher regions of speed of rotation have not yet be made. As M1 remains approx constant with a higher residence time, the Bo number decreases in inverse proportion to 5 or to the total number of revolutions as determined by ?. As M1 increases in the lower speed range with J??, Bo decreases in proportion to @ or to the square root of the total number of revolutions, as a result of the speed of rotation. Therefore, there is no uniform relation between longitudinal dispersion in continuous flow and the total number of revolutions, as has sometimes been supposed in the literature to be the case for (radial) batch mixing of solids in a rotating drum. 4.5
Special conditions
For a small inclination of 4” (Test 15) the Ml value of O-127 should be compared with a figure of O-087 cm’/sec, to be derived from Table 4 for a horizontal cylinder for the same working content. Axial mixing is measurably greater for an inclined cylinder. The Bo numbers calculated for increasing times in the test showed a tendency to decrease (from 38 to 26), indicating a somewhat different flow pattern. The introduction of small ribs into the drum (Experiments 10 and 11) gives Ml values of 0.110 and O-104, as compared with O-091 cm’/sec for Test 12, which had the same working content. The longitudinal mixing is only slightly increased. 4.6
Type of grain movement
For most tests rolling was ensured by the use of
II
emery paper. The smooth wall taken for Experiments 7 and 9 caused gliding and surging, with a low amplitude, on account of a low wall friction and high slip. The working content of Test 7 was abnormally low for the surging condition. Figure 2 shows the shape of the response concentration curves in comparision with two tests with good tumbling. The shape is quite different and for the Bo numbers a heavily decreasing trend with increasing t or F values was found, for Experiment 7 from 150 to 24 and for Experiment 9 from 110 to 25. The model for axial mixing is quite different for this type of particle motion. The flow pattern was completely different. A block of red rice could be seen to ascend very slowly up the wall and then to slide downwards. After some time some wedges of preponderantly red kernels were seen to be slowly moving along. The whole picture was reminiscent of SCHEIDEGGER’S [4] drunken man model, where mixing at a certain point or time depends on the prehistory of the local agglomeration. Visually, the impression was obtained of a high degree of axial mixing. However, the response curves and the very approximate Bo calculations show that longitudinal mixing is not greater for surging than for rolling and cascading. Of course, radial mixing is very poor. It may be remarked that the same tendency to find lower Bo values from the ascending branch of the response curve for longer times is found to a lesser degree for the smooth conical outlet shapes (Tests 30, 24, 25). Here gliding could be observed too. An inclination of the cylinder (Experiment 15) had the same effect on the Bo numbers. 4.7 Shape of inlet With a smoothly walled widely conical entrance shape (b), Experiments 22, 45 and 48 give an M1 figure of 0.046 cm2/sec, calculated with L = 50 cm. The effective length of the drum is decreased somewhat by the cone and therefore M1 is really somewhat lower still. For a rough cone (Test 47) 0.070 was calculated. Comparing these figures with 0.078 cm’/sec, derived from Table 4 for a straight entry face and the same working content, a lower overall axial mixing is observed, especially with the smooth conical entrance. The same difference is found on comparison of Test 60 (O-075 cm2/sec)
1095
R. RUTGERS to hold in this case too, calculation of the figure of 0.050 for Experiment 62 with G = 1300 g to 2600 g, as for Tests 22, 45 and 48, would also give J+f1= O-029 cm2/sec. The dispersion coefficient would be still lower for a lower speed of rotation.
with Tests 12 and 57 (O-092). The dead spot at the inlet side is largely removed and the local Bo number there is considerably increased. This means that the real axial mixing in the cylindrical part itself of a cylinder is even lower than would follow from the overall dispersion coefficients discussed above. 4.8
4.8.1
Dimension of a mixing cell
Using the simple formula of KR~MERS[5]:
Shape of exit
At the outlet too end-effects are to be expected. Eddies could be observed here. Some more or less conical pieces were added, as shown in Fig. 1. The effective length of the drum is thus increased and the Mi figures calculated for simplicity’s sake with L = 50 cm will be somewhat too low. With the weak cone No. 4 iV1 figures of 0.079 cm’/sec for a smooth exit and O-063 for a rough wall of the cone are found, as compared with 0.046 cm’/sec for a straight end face (orifice width of 6.5 cm for all cases). For the sharp cone No. 5 in Tests 30 and 31 with respectively a smooth and a rough cone wall, M1 is found to be O-140 and 0.071 cm’/sec. The orifice width was 9-O cm and a comparable figure for a straight end face with the same working content would be about 0.050 cm2/sec, assuming proportionality of M1 with fi (L value of 50 cm used). The conical-cylindrical outlet No. 6 with a smooth wall gives an M1 value of 0.22 cm2/sec or something like 0.27 when a greater length is used in the calculation. A comparable figure for a straight end face with the same rather low working content would be about 0.075 cm2/sec (Test 60). It is found that conical outlets with a smooth wall having a low wall friction coefficient cause a heavy local axial mixing effect and may considerably increase the overall dispersion coefficient. With rough cone walls the effect is much less, but still noticeable. Even vertical end-faces will have a disturbing influence. In section 4.2 it was stated that for a longer residence time M1 decreased from O-078 to 0.050 cm2/sec, corrected to a working content of 1300 g. Application of the same correction factor of 1.56 to the average figure of 0.046 cm2/sec for Experiments 22,45 and 48 with a content of 2560 g would result in a dispersion coefficient of 0.029. Assuming the proportionality of ikfr with l/@
1
-_=-=n - 1
2
2T.M,
Bo
L2
one finds with L = 50 cm, f = 600 set, for M1 values of, e.g., 0.3 and 0.05 cm2/sec figures for 12 of 8 and 43. The mixing cell lengths are then 6.3 and 1.16 cm. The latter figure for slower rolling and low axial mixing, as compared with the rice grain dimensions, the rice being orientated at random, would mean that the cell length is about 3 times the average particle dimensions. Axial mixing thus involves a grain and its two neighbours. 5.
DATA FROMTHE LITERATURE
No systematic investigations of the longitudinal mixing of solid particles flowing through rotating cylinders nor calculations of dispersion coefficients are known to the author. Some experiments are described in the literature and for some cases mixing coefficients were calculated by us from the data. RAOUF’S [6] measurements with fine sand (d, = 0.05 cm) in a horizontal cylinder of L = 126 cm and D, = 14 cm give for N = 10 rpm a coefficient of 0.23 cm2/sec. MISKELLand MARSHALL[7] described experiments with fine sand (d, = 0.05 cm) in a cylinder of L = 101 cm and D, = 13.9 cm, provided with 6 rather large flights. With N = 6 r.p.m. and volumetric degrees of filling from 0.1 to 0.01 ikfl values of 0.05 to 0.34 cm2/sec can be calculated from the data. HOGREBEand LEHMANN[8] did experiments with radioactive pulse feeding on an industrial rotary cement kiln for sludge feeding of 12,500 cm length and a diameter of 370 cm at the ends and 3 10 cm in the central section; the speed of rotation is not mentioned but was probably approx 1 r.p.m. From their cumulative exit concentration curves we
1096
Longitudinal
mixing of granular material flowing through a rotating cylinder-Part
calculate for Tests 2 and 3 Bodenstein numbers of 60 and 34 and dispersion coefficients of 173 and 320 cm2/sec. These are, however, overall figures and by far the most mixing takes place in the drain section and the sintering section. As it follows too from Hogrebe’s data on the response function at various distances in the kiln, the dispersion coefficient in the central smooth part where dry granules are moving must have been very much lower. SMITH[9] found and SAEMAN[lo] explained that the response curve with pulse feeding has two maxima for drums with rather large flights which take a lot of material; part of the material falls down before and another part past the centre of the cylinder. We did not find this phenomenon in our experiments without internal parts in the cylinder, nor with small ribs. With surging (see Fig. 2) smaller maxima may be found, as the coloured material may cling together rather persistently in groups. Literature about axial mixing in other types of flow of granular solids is still more scarce. For screw or paddle conveyors particle transport and longitudinal dispersion have been mentioned by PAWLOWSKI[l 11, VIERLINGand EPHREMUJIS[12], HUGGILL (see CREMER [13]), GREATHEADand SIMMONDS [141,RAOUF[6], and SCHULZE-PILLOT [151. Our own experiments [16] with a horizontal paddle conveyor of a size comparable with the rotating cylinder have given dispersion coefficients for long grain rice from 0.12 to 25 cm’/sec. At higher speeds of rotation the axial mixing increases very sharply. For a vertical riser with concurrent upward flow of gas and solid particles, conforming with pneumatic transport, VAN Zoom [17] determined extremely high axial dispersion coefficients of thousands cm21 set for the spheroidal granules of 20-150~ size. Only by using a long narrow tube, high velocities and very short residence times, could it be insured that the Bo number of the reactor would not be very low, e.g. 7. 6.
INFLUENCEOF PARTICLEPROPERTIES
With a round grain type of rice with kernel dimensions of 0.50 x 0.28 x 0.21 cm, an experiment (No. 17) with iV = 30 r.p.m. and G = 1395 g gave a dispersion coefficient of O-124, somewhat
II
higher than the comparable figure of O-099 cm2/sec for long rice at the same degree of filling of the cylinder. Longish particles may tend to interlock more, which decreases the rate of mixing. Further experiments with varying particZe shapes should be made. The effect ofparticle size itself has not been investigated. An indication of the influence of particle size distribution has been obtained by the (slug) feeding of a coloured mixture of 50% whole long rice (particle wt. O-0189 g) and 50% wt. broken rice (particle wt. 0.0074 g) in between white whole long rice. The response curve for the broken rice had a smaller width and a higher maximum compared with the whole grain. In the mixture the axial dispersion of the smaller particles was less than for the larger grains, the ratio of the coefficients being approx 1.6. Of course, the shape of the larger and the smaller particles was different in this experiment. This difference in dispersion is in agreement with the qualitative observation that caked twins of coloured rice grains, being larger than the individual kernels, come out of the cylinder later. The difference in the dispersion coefficients of smaller and larger particles in a mixture implies, for an experimental set-up as described with slug feeding, that at first relatively more larger particles will be found in the response with a relatively low total concentration of marked granules, followed later by more smaller particles. There will be a tendency for the Bo numbers calculated to decrease with time. This has been found to be conilrmed in our experiments. Another variable particle property is the degree of free-flowing or cohesion between the granules. Experiments have been made with long whole rice kernels rendered sticky by moistening with syrup. With sticky particles the top of the lunar crosssectional shape of the cylinder content become broader and the kernels are taken up higher. The head over the discharge opening was greater. In one case the grains were made moderately sticky and they formed balls and agglomerates on moving in the cylinder. During an experiment with N = 11 r.p.m. the agglomerates fell apart on rolling down and hitting the bottom of the cylinder. This disintegration caused greater axial displacements of the separated kernels and the diffusion coefficient was 5 times greater than with free-flowing rice grains.
1097
R. RUTGERS
In other experiments syrup was sprayed on the rolling grains at the beginning of the cylinder, making the particles wet-sticky. Agglomerates formed were loose, but on falling to the bottom they did not fall apart into their individual grains. The whole mass showed a certain coherence and the displacements of the solid particles relative to each other certainly appeared to be restricted. The dispersion coefficients ‘calculated were made comparable with those of free-flowing dry rice by correcting the latter for differences in working content and speed of rotation with the square root formulas. M1 values found for wet-sticky rice were equal to or even somewhat lower than for dry kernels. A number of experiments were made with cut out groats, which differ in several respects from the rice grains. The average kernel weight was 0.0046 g, the particle size distribution was much wider (sieve analysis from O-15 to 0.30 cm), the shape more angular and more equidimensional, and the surface properties were also different from those of rice. As would be expected for a non-homogeneous particle size there was a trend toward lower Bodenstein numbers where these were calculated with the aid of the Brenner tables for longer times of the ascending branch of the response curve. Calculation of Bo and M1 is either less accurate or more complex. This trend was much less marked for the higher speed of rotation of 60 r.p.m. The influence of the speed of rotation on the dispersion coefficient was as in the case of rice, approx proportional to J%, but the effect of varying the working content was less clear and more experiments are needed. The cut oat groats were used with moisture contents of 7 % and 25 %. In the latter case the moisture was distributed evenly throughout the kernels by tempering for several hours. The grains were not sticky but also free-flowing. The dispersion coefficient found was 1.1 times higher than that for the dry groats. On average, the M1 value for dry cut oat groats was 1.3 times higher than for long rice. 7.
INFLUENCE OF VARYING CYLINDER DIAMETER
This important subject needs much further elucidation and only some observations can be mentioned here, based on cut oat groats in cylinders
of 76.5 cm diameter. Consistent Bodenstein numbers could be calculated for the axial mixing in these larger drums. Table 7 gives some results for the larger and for the laboratory cylinders, with the fractions of the critical speed of rotation and the Froude numbers calculated. Table
7.
Axial dispersion of cut oat groats cylinders of differing diameters
No.
Dt (cd
L N N/No (cm> (wm)
I II
76.5 76.5
230 149
15 0.31 19 0.39
O+IO97 044 0.155 0.084
E
16.0
50
60 30 0.57 0.28
0.32 0.080
C
16.0 50 30 0.28 0.080 0.18
Fr
P
0.20 0.33
Moisture content groats (%) 7 25
7
25
in
MI (cd/ SW)
0.7
1.0 0.085 0.11
0.085
As a very approximate first orientation of the diameter effect, these experiments would indicate that for a 4.8 fold increase in D,, the dispersion coefficient increases by 8.2 x or, supposing proportionality to exist with (DJDJ’, n would be only 1.34. The effect of larger diameters would be relatively small and smaller than the capacity increase with wider reactors. 8.
CONCLUSIONS
To obtain lower longitudinal dispersion, given a certain average residence time, it is advisable in the case of rotating cylinders to use low rotation speeds, high loadings, a horizontal cylinder position, no internal ribs, or rather small ones, a conical smooth entry end face and a straight outlet end face. If made roughly walled, a moderately conical outlet does not cause much more axial mixing. Surging of the particles, as compared with rolling and tumbling, gives a different mixing pattern but no greater amount of overall axial dispersion. When the reaction to be carried out involves only a reaction between solid and gas, the gas being in excess and penetrating rapidly into the interstitial voids, surging and gliding in the cylinder gives no disadvantage from the viewpoint of longitudinal dispersion. When liquid is sprayed in, however,
1098
Longitudinal
mixing of granular material flowing through a rotating cylinder-Part
good tumbling and radial mixing is required and surging must be avoided. For cohesive particles forming agglomerates and rolling down as agglomerates which burst asunder on falling on the bottom of the cylinder, axial dispersion may increase several fold. For wetsticky particles which keep their place in the rotating mass, the longitudinal dispersion coefficient may, however, be even lower as compared with dry free-flowing granules. Given a certain residence time and dispersion coefficient, the axial mixing can be reduced sharply by increasing the length of the cylinder according to the Bodenstein formula (5). For greater length, the diameter may be reduced, which lowers the M1 value and thus further increases the Bo number. Although several regularities with regard for example to the influence of the degree of filling and of the speed of rotation have been discussed in the text, it is felt that further experiments and data are needed, especially in respect of the diameter effect, before an attempt is made to give a generalized picture of longitudinal dispersion for the present type of particle flow. The problems of scaling up and down and of the dynamic similarity and of optimum dimensions of a rotating cylinder can then be handled. A Bodenstein number of 50 for flow in a reactor is generally regarded as meaning very limited axial mixing. According to Equation (4) however, the variance is then O-039 or the standard deviation 0.20. A a-figure of 0.20 would mean, of course, that 32% of the particles would have a residence time deviating by more than 20% from the mean time and 5% of the grains would deviate by more than 40%. Whether this is to be judged a high or a low variability depends on the type of reaction and the required specifications for the product. It will not always be negligible. The axial dispersion coefficient for granular material flowing through a rotating cylinder can be very low indeed and compares favourably with the coefficients mentioned in the literature for other types of flow of fluids. HOFFMANN[ 181and FRAMERS and WESTERTERP [19] have given surveys of the subject, e.g. flow of liquids and gases through tubes and annuli, flow of liquids through a rotating disc
II
reactor, fluid flow through packed and fluidized beds of solid particles. It may be said that solid particles flowing through a rotating cylinder are semi-fluidized radially by mechanical means, resulting in a rather dense phase, at least at lower speeds of rotation. Longitudinal dispersion can be made smaller than radial mixing. By choosing appropriate dimensions and operating conditions for the rotating cylinder rather large Bodenstein numbers for the process can be obtained, and this type of reactor might deserve more attention in the chemical industry [20]. Acknowledgements-Thanks are due to Mr. H. A. BOK, Managing Director of Royal Lassie Mills Ltd, for permitting publication of this work and to Mr. W. Ev~as for carefully carrying out the experiments and for his considerable further help. Discussions on the subject with Mr. W. A. BEVERLOO and, especially, with Dr. A. KLINKENBERG and Mr. S. J. VELLINGA. who allowed their method of calculation for slug-feeding to be used, helped me to clarify my present ation of the matter.
NOTATION Bodenstein number g Concentration, fractional Cylinder diameter, cm Particle diameter, cm Concentration, fractional (step-function) Working content, g Acceleration due to gravity, cm/seca Length of cylinder, cm Longitudinal dispersion coefficient, cma/sec Radial dispersion coefficient, cmz/sec Speed of rotation, r.p.m. and r.p.s., secl Critical speed of rotation, secl Number of ideal mixers in a cascade Fractional volume, degree of filling (fractional loading) Mass flow rate, g/set Radius of cylinder, cm Radial position of a particle in a cylinder, cm Reynolds number Variance of response curve Time, set Feeding time of coloured grain, set Average residence time, including &t,, set Linear velocity, cmjsec Peripheral velocity, cm/set Distance, cm Angle of internal friction Inclination of cylinder Angle of repose Porosity of collection of solid particles Relative time I/T Relative time of feeding marked particles Wall friction coefficient
R. RUTGERS 7 ? TH w
P Specific density, g/cm3 PB Bulk density, g/cm3 Standard deviation 0: Variance
Residence time, set Average residence time, set Half value time, set Angular speed, set-l
REFERENCES BRENNERH., Chem. Engng. Sci. 17 229 1962. VAN DEEMTERJ., ZIJIDERWEG F. J. and KLINKENBERG,A., Chem. Engng. Sci. 5 271 1956. KLINKENBERGA., Trans. Inst. Chem. Engrs. London, 43 T141 1965. ScrrsmrsooER A. E., Physics of Flow through Porous Media, Toronto 1960. KRAMERSH. and ALBERDAG., Chem. Ewvtg. Sci. 2 173 1953. RAOUF M. S., Continuous Mixing of solids. Diss. Wageningen 1963. MISKELLF. and MARSHALLW. R.. Chem. Enona. Proar. 52 355 1956. H~GREBEK. and LEHMANN W. S.,‘Zement KGk-Gips.‘b 133,210 1956. SMITHB. A., Trans. Am. Inst. Chem. Engrs. 38 251 1942. SAEMANW. C. and MITCHELLT. R., Chem. Engng. Progr. 50 467 1954. PAWLOWSKIJ., Chem. Zna. Tech. 26 699 1954. VIERLXNGA. and &HRE&S CH., Fiirdern und Heben 7 433, 490 1957. CREMERH. W. and DAVIEST., (Editors). Chemical Erzaineerina Practice. Volume 3. Solid Systems, London 1957. GREATHEADJ. A A. and S&&NDS W.‘IL C., Chem. &gng. >rogr. 53 i94 1957 _ SCHULZE-PILU)TG., Paper read at Achema Congress, Frankfurt 1964. RUTGERSR., Unpublished data. VANZ~~NEN D., Third Congress of European Federation of Chemical Engineers, London 1962, A54. HO~NN H., Chem. Eng&. Sci.-14 193 1961. K. R.. Elements of Chemical Reactor Desian and Ooeration. Amsterdam 1963. ‘,:i; KRAMEas H. and WESTERTERP PO1 MCCORMICKP. Y., in PERRYJ. H. (Editor). Ch.emical Engineers Handbook 20 15.-New York 1963. HI PI r31 [41 151 161 [71 PI 191 m r111 r121 1131 r141 1151 [I61 I171
R&m&-L’auteur prQente des experiences sur la dispersion axiale dans un Ccoulement continu, par exemple, de grains de riz dans un cylindre rotatif ii l’echelle du labo:atoire. 11 montre la validit de son modele d’ecoulement qui comporte la possibilite d’imposer une diffusivite longitudinale. Les coefficients de dispersion axiale ont Cte d&ermines sous diverses conditions: remplissage, vitesse de rotation, nombre de rotations, debit, inclinaison, mouvement des particules et formes de Pent& et de la sortie du cylindre. De plus, les effets de la variation de quelques proprietb des particules solides ont BtCetudies, par exemple, la forme des particules, la distribution des diverses tailles de particules, et enfIn la cohesion entre particules. Quelques experiences ont &tCfaites avec un cylindre de plus grand diametre. Les coefficients de dispersion longitudinale ont pu etre pris assaz petits, et inferieurs par exemple a 0,04 cm2/sec. tres inferieurs done a ceux obtenus avec la plupart des autres types d’appareils avec ecoulement de fluides. Zusarnmenfasstmg-Es werden Versuche beschrieben, bei denen langkiirniger Reis kontinuierlich durch einen rotierenden Experimentier-Zylinder Ilie&, urn dessen axiale Verteilung zu untersuchen. Eine Modellvorstellung, bei der die Pfropfenstriimung von axialer Riickvermischung tiberlagert wird, erwies sich als gtiltig. Axiale Mischkoefbzienten wurden unter verschiedenen Bedingungen bestimmt, wobei der Durchsatz, die Rotationsgeschwindigkeit, die lineare Stromungsgeschwindigkeit, die Zylinderneigung, die Art der transportierten Partikel und die Gestaltung der zylindrischen Eintrittsbzw. Austrittsoffnungen verandert wurden. Dariiber hinaus wurden einige Partikeleigenschaften verandert, wie z.B. ihre Form, ihre KorngriiBenverteilung und ihre gegenseitige Haftung. Auch von einigen Experimenten bei vergriigertem Zylinderdurchmesser wird berichtet. Die axialen Mischkoeffizienten kijnnen sehr klein sein (herab bis 0,04 cma/sec) und liegen damit niedriger als in den meisten Stromungsprozessen und Reaktoren.
1100
Pergamon Press Ltd., Oxford.
Printed in Great Britain.
Longitudinal mixing of granular material flowing through a rotating cylinder Part II. Experimental R. RUTGERS Laboratory
Royal Lassie Mills Ltd., Wormerveer,
the Netherlands
(Received 3 February 1965) Abstract-Experiments on axial dispersion for continuous flow of, e.g. long grain rice through a laboratory-scale rotating cylinder are reported. The model of piston flow with superimposed longitudinal diffusivity proved to be valid. Axial dispersion coefficients were determined under varying conditions of loading, speed of rotation and number of rotations, linear flow velocity, inclination, types of particle movement, shapes of cylinder entrance and exit end faces. Furthermore, the effect of some variations in the properties of the solid particles has been studied e.g. particle shape, particle size distribution and particle cohesion. Some experiments with a cylinder of larger diameter are reported. Longitudinal dispersion coefficients could be rather low, e.g. down to 0.04 cm2/sec, much lower than most other types of fluid flow processes and reactors.
1. EXPERIMENTS EXPERIMENTS are described for a small laboratory
scale cylinder of perspex with L = 50 cm and D, = 16.0 cm (NC = 106 r.p.m.), with varying dimensions and shapes of the inlet and outlet sections, involving variation in the working content (see Fig. 1). The straight discharge end faces were made of aluminium and all other end faces of Perspex, sometimes covered with emery paper. For most tests the inner wall of the cylinder was covered with emery paper, to prevent slip. In some tests the smooth wall was used and surging took place. The speed of rotation was varied between 8 and 97 r.p.m. or O-133 and l-62 r.p.s. In one test the drum was given an inclination of 4”; for the other experiments the cylinder had a horizontal position. With some tests 4 smaller wooden radial ribs, extending over the entire length, of 0.6 cm thickness and 05 or
FIG. 1. Experimental
1.2 cm height were inserted, equally spaced over the circumference. In some tests the effective length of the cylinder was halved. As granular material, mostly air-dry free-flowing long-grain milled rice with a moisture content of 13 % was used, with particle dimensions of O-70 x 0.21 x 0.17 cm. The average kernel weight was O-0188 g with a standard deviation of only 0.0017 g. Some tests were performed with sticky rice and with a mixture of whole and broken kernels. As another granular solid the more angular cut oat groat particles were used. Some of the grains were coloured red with fuchsine, care being taken not to change their surface properties. All tests were done with continuous flow and with vibrator feeding. The feeding rate could vary somewhat during a test, e.g. +5 ‘A. After reaching a steady state with white rice, red rice was fed, for practical reasons for about O-4 of the average
drum with inlet and discharge variations.
1089
R. RUTGERS residence time, followed again by white rice. The residence time used was mostly about 600 set, with some variations. The working content was checked repeatedly. At the outlet samples of about 5-20 g were taken for about 5-10 set, with a time interval of 15-30 sec. This method of sampling would seem to approach point sampling near enough to render it unnecessary to introduce an extra variance. For a number of experiments the whole response curve was determined, whereas for the other tests only the rising branch of the exit concentration curve up to a concentration of 0.5 was determined. The content of red grain was determined by visual selection by hand and expressed as a weight fraction, which in this case is identical with a volume or number fraction. Recovery of red grains was satisfactory. No serious contamination of the white kernels took place.
2. CALCULATIONS The exit concentrations of red-coloured grains were put in t-F response curves at the average time of the duration of the sampling. Smooth lines could be drawn through the points. As for most tests red kernels began to appear after about +? and red rice was fed for O-42, the exit concentration should be usable up to about 0*85i (up to F-values of e.g. 0.35) for the step-function calculation of the Bodenstein numbers. The exit concentration tables of BRENNER [I], based on finite vessel length, were used for deriving Bofigures from the experiments. Taking 3 or 4 points of the ascending branch of the response curve gave Bo-figures which were reasonably constant in most cases. This confirms the validity of the chosen diffusivity model for the present type of flow of granular material. The average spread from the mean Bo-value was + 5 %. There were some exceptions to the constancy of the Brenner Bo-figures, which will be discussed later on (Tests 7, 9, 15, 24 and 25). For a number of experiments Bo values were calculated with the Danckwerts formula (See Part I). The results were 20-15 % lower, in the range of Bo from 20-100, than the Brenner results, and there was a definite trend towards lower Danckwerts Bo’s when calculations were made for longer times in a
test. This model is not the most appropriate for our case, even for higher Bo numbers. The total exit concentration curves has the familiar bell shape with a tendency to tailing. Two examples, Nos. 13 and 16 are given in Fig. 2. The different shape for Tests 7 and 9, which indicated a different flow pattern of the grains, will be discussed later on. The cumulative concentration curve on a probablility scale against time on a logarithmical scale has been shown for some examples in Fig 3. Up to 90-95x rather straight lines are obtained, indicating a log-normal distribution, especially for higher Bo numbers. Applying the criterion of VAN DEEMTERet al. [2]. for a feeding time of marked material which is possible without changing the width of the response curve, we find O*lf for a Bo number of 50 (using mixing cell length as the theoretical plate length). It is seen that a feeding time for coloured grain of 0.47 is too long to allow the use of pulse function formulas for the calculation of Bo numbers. KLINKENBERG [3] suggested statistical treatment be applied for the calculation of the Bo number in the case of slug feeding, using the whole response curve. The distribution curve is transposed into the cumulative curve, using $-min intervals. By numerical partial integration according to SIMPSON with I-min intervals, the average residence time T including half of the feeding time t, of coloured grain is found:
7 = -
G
Qlll
is the average residence
time
of grains
in
the cylinder. By numerical integration of t( 1 - F) the variance of the concentration-time curve is found from : (1 - F)t dt - T’)
(2)
S2 is the sum of the relative variance a2 of the residence time of the grains in the rotating cylinder and the variance of the feeding time of coloured particles,
1090
which is z: s2 = 02 +
2
(3)
Longitudinal
mixing of granular material flowing through a rotating cylinder-part
t FIG. 2.
-$- 1 -
$0” - $(02)2 - gr’)”
see
Typical response curves.
From c2 the Bodenstein number can be calculated according to Formula (2) in Part I, which is made easier for calculations by Klinkenberg in the following transformation: Bo =
II
(4)
This calculation was carried out for a dozen experiments. The calculated feeding times of coloured grains was on average a fraction of 0.83 of the actual times used. The Bo values were 0.89 times the numbers calculated from the ascending branch of the distribution response curve with the use of the Brenner tables. With regard to the validity of the models used as a basis for calculation of 02, of Bo from Formula (4) and of the dispersion coefficient M1 it should be borne in mind that some irregularities are apt to occur at the entrance and at the exit of the cylinders. With vibrator feeding the change from normal grain to coloured kernels and back to white again involves some variations in feeding velocity. At the entry coloured grains tend to be fed over a certain length of the cylinder and not at a point,
and furthermore it takes some time to distribute them evenly over the cross-section by radial mixing. At the outlet the grains strike against the vertical end-face, which pushes them back, causing swirls and eddies which increase local axial mixing. The tail of the response curve will be made longer. Furthermore the grains must leave the cylinder through a narrow opening under the pressure of a considerable head, whilst the particles are moving radially. The effects of various forms of the endfaces, as shown in Fig. 1, have been investigated and will be discussed later on. Most experiments were done by measuring only the first part of the response curve, and the Bo numbers used for calculation of the dispersion coefficients in cm2/sec, according to Formula (5), were derived from the Brenner tables.
3. &SXJLTS In Table 1 the cumulative response curve figures are given for a number of experiments. Table 2
1091
R. RUTGERS
360 240
-13 j-5
63 13.4 15 /0 0
x-__---_,6
120
o__--_- 7 ~________ cJ 0001
0.010
0.100 0.200
0.500
0.8000.900
0.990
0,999
F FIG. 3.
Table 1. 6
0400
oaoo
O-000
0.000
0900
OMO
0400
0400
O-000
0400
0.000
0400 0400 0400
0.002 0.009 0.026
0900 0400
0400 0900
6 7 8 9 10
0.001 0.023 0.088 0,195 0.348
0.060 0.121 0.217 0.345 0.471
0.000 0300 0.006 0.046 0.158
11 12 13 14 15
0.535 0.712 0.839 0.915 0.956
0.583 0.677 0.753 0.814 0.862
16 17 18 19 20
0.979 0.991 0.996 0.999 -
21 22 23 24 25
-
Time (min) 0
2
Cumulative response concentrations
5
Test No.
o*ooo
Cumulative response curve.
11
12
13
14
0400
0.000
omo
0.000
0.000
0400
0400
0.000
o*ooo
0400
O-000
OGOO
oaoo
o*ooo 0900 omo o*ooo
0000
0400
0.000 0000 0.000
0400 0400 0.000 0400
OGOl OW8 0.038 0.107 0.213
0.002 0.016 0.056 0.139 0.265
0*004 0.024 0.082 0.192 0.354
0.001 0.011 0.045 0,124 0.249
0.352 0.583 0.763 0.875 0.931
0.345 0,496 0.637 0.756 0.847
0.431 0.602 0.747 0.848 0.916
0.549 0.729 0.853 0.917 0.954
0.897 0.923 0.945 0.961 0.972
0.959 0.977 0.987 0.992 0.995
0.911 0.951 0.974 0.986 0.994
0.955 0.977 0.987 0.993 0.997
0.975 0.987 0.994 0.997 0,999
0.979 0.985 0.990 0.993 0.996
0.997 -
0.997 0.998 -
0.998 0.999 -
o*ooo
8
omo
10
o*ooo omo oaoo
-
1092
16
17
0400
0.000
OGOO
0400
0900
0400 0400
0400 0300
0400 O+lOl
0400 OGOO 0400 0300 0.003
0.003 0.023 0.089 0.226 0.420
0400 0.003 0.024 0.081 0.191
0.005 0.033 0.112 0.255 0447
0.017 0.053 0.120 0.213 0.327
0.002 oGO9 0,035 0.087 0.165
0.419 0.592 0.732 0.835 0.903
0.634 0.804 0.904 0.954 0.974
0.349 0.542 0.719 0.851 0.926
0.627 0.762 0.854 0.912 0.949
0.452 0.571 0.675 0.763 0.832
0.270 0405 0.552 0.690 0.800
0.946 0.971 0.984 0.990 0.994
0.986 0.994 0.997 0.999 -
0.965 @984 0.993 0.997 0.999
0.972 0.985 0.992 0.996 0.998
0.883 0.921 0.947 0.963 0.974
0.877 0.922 0.951 0.970 0.982
0.997 0.998 -
-
-
-
0.980 0.985 0.989 0.992 0.995
0.988 0.992 0.995 0.996 0.997
-
1.5
oaoo 0400 0400
Longitudinal
mixing of granular material flowing through a rotating cylinder-Part
gives details of these tests together with those of other experiments with the same type of long dry whole rice kernels; Bo and MI figures are mentioned. For the indications of various entry and discharge openings, see Fig. 1. Furthermore the speed of rotation in r.p.s., the working content G in grams, the feed rate Q, in g per set, the average residence time i in sets and the time of feeding coloured rice 0, as a fraction of 7 at the same flow rate are recorded. The dispersion coefficients are calculated
Table 2.
II
for experiments with conical end-faces with L = 50 cm (see below). The fractional volumetric degree of filling P of the cylinder was approx. O-15 for a content G of 1000 g. This factor was somewhat lower for higher loading and also lower when surging occurs (Experiments 7 and 9). Higher speed of rotation, causing lower dynamic bulk density, increases P (Experiments 8, 16). Further experiments with grains of varying particle properties will be discussed later on.
Experimental results -
No.
Inter
Outlet
14 13 32 12 2 57
r.p.s.
Variations
G(g)
Qm 63/w)
C (set)
0,
Bo
MI
(cm2/sec)
0.50 0.50 0.50 o-50 0.50 0.50
-
4010 2620 2750 1390 1760 1575
6.55 4.85 4.32 2.32 3.15 2.37
612 540 637 600 559 591
0.38 0.41 040 0.39 0.38 0.39
:: 47 46
0.064 0.074 0.083 0.091 0.095 0.092
63 63
6 8 58 16
a a a a
3 3 3 3
0.133 1.00 l*OO 1.62
-
1620 1635 1850 2490
2.50 2.37 3.17 4.03
648 613 584 612
0.37 040 0.41 0.39
89 44 33 15.0
0.044 0.093 0.129 0.270
15 :1”
a a a
1 3 3
0.50 0.50 0.50
CL< = 4” ribs, 0.4 cm ribs, 1.2 cm
2030 1340 1305
3.60 2.23 2.28
564 600 572
0.40 0.41 040
35 38 42
0.127 0.110 0.104
4 5 7 9
a a a a
0.50 0.50 0.50 0.133
L=25cm L=25cm smooth wall smooth wall
770 730 1510 3300
2.30 1.37 2.47 2.38
335 548 607 1385
0.85 0.31 040 0.27
20 13.4 ? ?
0.093 0.085 ? ?
22 :: 47 2 61 62 23 46 30 ;: 24
b, sm b, sm b, sm b, r b, sm b, sm b, sm b, sm b, b, b, b, b, b,
sm sm sm sm sm sm
4, sm 4, r 5, sm 5, r 6, sm 6, sm
0.50 0.50 o-50 0.50 0.50 0.50 0.50 0.50
-
2595 2580 2520 2550 1600 1440 1325 1300
4.33 4.50 4.08 4.17 5.47 2.25 1.07 0.52
599 573 617 612 294 640 1284 2516
0.39 040 040 040 0.80 0.41 0.29 0.26
105 98 76 58 120 52 34 20
0.041 0.045 0.053 0.070 0.071 0.075 0.058 0.050
050 0.50 0.50 0.50 0.50 0.50
-
2510 2640 2150 2270 1425 1385
435 4.35 3.98 3.78 2.27 2.22
577 606 540 600 629 625
040 040 0.40 0.41 0.39 0.55
55 65 33 59 18 23
0.079 0.063 0.140 0.071 0.220 0.174
sm = smooth (Perspex.) Ml figures from No. 22 on are calculated with L = 50 cm (see text). r = rough (emery paper.)
1093
R. RUTGERS
4. 4.1
the dispersion coefficient to I/ JF or l/ JG.
DISCUSSION OF THE RESULTS
Reproducibility
Table 3 contains some data for M1 on the repetition of experiments, which give an idea of the accuracy of the determination and the calculation of the dispersion coefficient in the way which has been described. Table 3.
Reproducibility Ml
A
13 32
0.074 0.083 >
0.009
2 57 12
0.095 0.091I0,091
0.004
8 58
0.093 0.129 >
0.036
45 48 22
0045 0.053> 0.041
0.012
25 24
0.220 0174 >
0.046
2, 12, 57 5 59 60 61 62
2,12,57 13,32 14
a
Outlet b, smooth
b” :: b
4.4
1575 583 730 548 1600 294 1440 640 1325 1284 1300 2516
0.086 0.046 0.170 0.078 0.039 O-020
0.093 0.085 0.071 0.075 0.058 0.050
Speed of rotation and number of revolutions
The results for experiments with an increasing speed of rotation at constant residence time and the same inlet a and outlet 3 are collected in Table 6. Here the ratio of N to the critical speed of rotation NC, the ratio M,/&?(N in r.p.s.)‘and the Froude W2R
Influence of loading
0.25 0.41 0.58
u MI (cm/set) (cm+ec)
values for a greater length and also probably for a longer residence time may result from the relatively smaller influence of the end effects on the overall Bo and MI figures and is indicated in the M1 figures found. However, the somewhat lower working content at a lower feed rate will have had the opposite effect on M1.
number -
1575 2685 4010
50 25 50 50 50 50
t (set)
For a range of u between 0.02 and 0.17 cm/set there is only some variation in the dispersion coefficient. The tendency to obtain lower M1
mentioned.
P
L (cm) G(g)
N = 60 r.p.m
The degree of volumetric filling P has been calculated from the working content G and the dynamic bulk density, which varies slightly with the working content. The results for tests with 30 r.p.m. are given in Table 4, where the product M,*,/P is also
G(g)
Influence of axial linear velocity Inlet
Test
Remarks
Loading G, degree ofJilling P
Test
Axial linear velocity
Table 5.
Somewhat greater variation is observed at a speed of rotation of 60 r.p.m. in the transition region prior to cataracting and with the smooth longish conical outlet 6, which gave more gliding, backswirling and local axial mixing.
Table 4.
proportional
Table 5 gives the results for some tests with variation of length and residence time, all for N = 30 r.p.m. and outlet No. 3.
of experiments
Test No.
4.2
4.3
is approx.
Ml
are also given.
Table 6.
Ml4
(cm2/sec)
0.093 0.078 0.064
9
No.
0.047 0.050 0.049
It is seen that for the chosen moderate speed of rotation with the rolling type of particle motion,
6 2,12,57 8,58 16
1094
N
Influence of speed of rotation N
N
+= G
MI
(r.p.s.) (r.p.m.) NC (s)
(g)
0.133 0.50 1.00 1.62
1620 1575 1740 2490
8 30 60 97
0.075 0.28 0.57 0.92
648 583 598 612
MI
Froude
(cmQec) ~‘8 number 0.044 0.093 O-111 0.270
0.12 0.13 O-11 0.21
OT1057 0.080 0.32 0.84
Longitudinal
mixing of granular material flowing through a rotating cylinder-Part
The working content was rather constant up to 60 r.p.m. and only minor corrections of Mr for varying content would be needed. Under cataracting conditions G was considerably higher. For a lower content M1 would be higher still, but it has not been investigated whether a proportionality with l/,/G holds under these conditions too. Up to about 60 r.p.m., the speed of rotation region where cataracting started for this type of grain, M1 increases in proportion to approx J??. At a higher speed a much sharper increase of the dispersion coefficient is found. Close to N, a decrease of M1 might occur. Further investigations into the relation between M1 and N in the higher regions of speed of rotation have not yet be made. As M1 remains approx constant with a higher residence time, the Bo number decreases in inverse proportion to 5 or to the total number of revolutions as determined by ?. As M1 increases in the lower speed range with J??, Bo decreases in proportion to @ or to the square root of the total number of revolutions, as a result of the speed of rotation. Therefore, there is no uniform relation between longitudinal dispersion in continuous flow and the total number of revolutions, as has sometimes been supposed in the literature to be the case for (radial) batch mixing of solids in a rotating drum. 4.5
Special conditions
For a small inclination of 4” (Test 15) the Ml value of O-127 should be compared with a figure of O-087 cm’/sec, to be derived from Table 4 for a horizontal cylinder for the same working content. Axial mixing is measurably greater for an inclined cylinder. The Bo numbers calculated for increasing times in the test showed a tendency to decrease (from 38 to 26), indicating a somewhat different flow pattern. The introduction of small ribs into the drum (Experiments 10 and 11) gives Ml values of 0.110 and O-104, as compared with O-091 cm’/sec for Test 12, which had the same working content. The longitudinal mixing is only slightly increased. 4.6
Type of grain movement
For most tests rolling was ensured by the use of
II
emery paper. The smooth wall taken for Experiments 7 and 9 caused gliding and surging, with a low amplitude, on account of a low wall friction and high slip. The working content of Test 7 was abnormally low for the surging condition. Figure 2 shows the shape of the response concentration curves in comparision with two tests with good tumbling. The shape is quite different and for the Bo numbers a heavily decreasing trend with increasing t or F values was found, for Experiment 7 from 150 to 24 and for Experiment 9 from 110 to 25. The model for axial mixing is quite different for this type of particle motion. The flow pattern was completely different. A block of red rice could be seen to ascend very slowly up the wall and then to slide downwards. After some time some wedges of preponderantly red kernels were seen to be slowly moving along. The whole picture was reminiscent of SCHEIDEGGER’S [4] drunken man model, where mixing at a certain point or time depends on the prehistory of the local agglomeration. Visually, the impression was obtained of a high degree of axial mixing. However, the response curves and the very approximate Bo calculations show that longitudinal mixing is not greater for surging than for rolling and cascading. Of course, radial mixing is very poor. It may be remarked that the same tendency to find lower Bo values from the ascending branch of the response curve for longer times is found to a lesser degree for the smooth conical outlet shapes (Tests 30, 24, 25). Here gliding could be observed too. An inclination of the cylinder (Experiment 15) had the same effect on the Bo numbers. 4.7 Shape of inlet With a smoothly walled widely conical entrance shape (b), Experiments 22, 45 and 48 give an M1 figure of 0.046 cm2/sec, calculated with L = 50 cm. The effective length of the drum is decreased somewhat by the cone and therefore M1 is really somewhat lower still. For a rough cone (Test 47) 0.070 was calculated. Comparing these figures with 0.078 cm’/sec, derived from Table 4 for a straight entry face and the same working content, a lower overall axial mixing is observed, especially with the smooth conical entrance. The same difference is found on comparison of Test 60 (O-075 cm2/sec)
1095
R. RUTGERS to hold in this case too, calculation of the figure of 0.050 for Experiment 62 with G = 1300 g to 2600 g, as for Tests 22, 45 and 48, would also give J+f1= O-029 cm2/sec. The dispersion coefficient would be still lower for a lower speed of rotation.
with Tests 12 and 57 (O-092). The dead spot at the inlet side is largely removed and the local Bo number there is considerably increased. This means that the real axial mixing in the cylindrical part itself of a cylinder is even lower than would follow from the overall dispersion coefficients discussed above. 4.8
4.8.1
Dimension of a mixing cell
Using the simple formula of KR~MERS[5]:
Shape of exit
At the outlet too end-effects are to be expected. Eddies could be observed here. Some more or less conical pieces were added, as shown in Fig. 1. The effective length of the drum is thus increased and the Mi figures calculated for simplicity’s sake with L = 50 cm will be somewhat too low. With the weak cone No. 4 iV1 figures of 0.079 cm’/sec for a smooth exit and O-063 for a rough wall of the cone are found, as compared with 0.046 cm’/sec for a straight end face (orifice width of 6.5 cm for all cases). For the sharp cone No. 5 in Tests 30 and 31 with respectively a smooth and a rough cone wall, M1 is found to be O-140 and 0.071 cm’/sec. The orifice width was 9-O cm and a comparable figure for a straight end face with the same working content would be about 0.050 cm2/sec, assuming proportionality of M1 with fi (L value of 50 cm used). The conical-cylindrical outlet No. 6 with a smooth wall gives an M1 value of 0.22 cm2/sec or something like 0.27 when a greater length is used in the calculation. A comparable figure for a straight end face with the same rather low working content would be about 0.075 cm2/sec (Test 60). It is found that conical outlets with a smooth wall having a low wall friction coefficient cause a heavy local axial mixing effect and may considerably increase the overall dispersion coefficient. With rough cone walls the effect is much less, but still noticeable. Even vertical end-faces will have a disturbing influence. In section 4.2 it was stated that for a longer residence time M1 decreased from O-078 to 0.050 cm2/sec, corrected to a working content of 1300 g. Application of the same correction factor of 1.56 to the average figure of 0.046 cm2/sec for Experiments 22,45 and 48 with a content of 2560 g would result in a dispersion coefficient of 0.029. Assuming the proportionality of ikfr with l/@
1
-_=-=n - 1
2
2T.M,
Bo
L2
one finds with L = 50 cm, f = 600 set, for M1 values of, e.g., 0.3 and 0.05 cm2/sec figures for 12 of 8 and 43. The mixing cell lengths are then 6.3 and 1.16 cm. The latter figure for slower rolling and low axial mixing, as compared with the rice grain dimensions, the rice being orientated at random, would mean that the cell length is about 3 times the average particle dimensions. Axial mixing thus involves a grain and its two neighbours. 5.
DATA FROMTHE LITERATURE
No systematic investigations of the longitudinal mixing of solid particles flowing through rotating cylinders nor calculations of dispersion coefficients are known to the author. Some experiments are described in the literature and for some cases mixing coefficients were calculated by us from the data. RAOUF’S [6] measurements with fine sand (d, = 0.05 cm) in a horizontal cylinder of L = 126 cm and D, = 14 cm give for N = 10 rpm a coefficient of 0.23 cm2/sec. MISKELLand MARSHALL[7] described experiments with fine sand (d, = 0.05 cm) in a cylinder of L = 101 cm and D, = 13.9 cm, provided with 6 rather large flights. With N = 6 r.p.m. and volumetric degrees of filling from 0.1 to 0.01 ikfl values of 0.05 to 0.34 cm2/sec can be calculated from the data. HOGREBEand LEHMANN[8] did experiments with radioactive pulse feeding on an industrial rotary cement kiln for sludge feeding of 12,500 cm length and a diameter of 370 cm at the ends and 3 10 cm in the central section; the speed of rotation is not mentioned but was probably approx 1 r.p.m. From their cumulative exit concentration curves we
1096
Longitudinal
mixing of granular material flowing through a rotating cylinder-Part
calculate for Tests 2 and 3 Bodenstein numbers of 60 and 34 and dispersion coefficients of 173 and 320 cm2/sec. These are, however, overall figures and by far the most mixing takes place in the drain section and the sintering section. As it follows too from Hogrebe’s data on the response function at various distances in the kiln, the dispersion coefficient in the central smooth part where dry granules are moving must have been very much lower. SMITH[9] found and SAEMAN[lo] explained that the response curve with pulse feeding has two maxima for drums with rather large flights which take a lot of material; part of the material falls down before and another part past the centre of the cylinder. We did not find this phenomenon in our experiments without internal parts in the cylinder, nor with small ribs. With surging (see Fig. 2) smaller maxima may be found, as the coloured material may cling together rather persistently in groups. Literature about axial mixing in other types of flow of granular solids is still more scarce. For screw or paddle conveyors particle transport and longitudinal dispersion have been mentioned by PAWLOWSKI[l 11, VIERLINGand EPHREMUJIS[12], HUGGILL (see CREMER [13]), GREATHEADand SIMMONDS [141,RAOUF[6], and SCHULZE-PILLOT [151. Our own experiments [16] with a horizontal paddle conveyor of a size comparable with the rotating cylinder have given dispersion coefficients for long grain rice from 0.12 to 25 cm’/sec. At higher speeds of rotation the axial mixing increases very sharply. For a vertical riser with concurrent upward flow of gas and solid particles, conforming with pneumatic transport, VAN Zoom [17] determined extremely high axial dispersion coefficients of thousands cm21 set for the spheroidal granules of 20-150~ size. Only by using a long narrow tube, high velocities and very short residence times, could it be insured that the Bo number of the reactor would not be very low, e.g. 7. 6.
INFLUENCEOF PARTICLEPROPERTIES
With a round grain type of rice with kernel dimensions of 0.50 x 0.28 x 0.21 cm, an experiment (No. 17) with iV = 30 r.p.m. and G = 1395 g gave a dispersion coefficient of O-124, somewhat
II
higher than the comparable figure of O-099 cm2/sec for long rice at the same degree of filling of the cylinder. Longish particles may tend to interlock more, which decreases the rate of mixing. Further experiments with varying particZe shapes should be made. The effect ofparticle size itself has not been investigated. An indication of the influence of particle size distribution has been obtained by the (slug) feeding of a coloured mixture of 50% whole long rice (particle wt. O-0189 g) and 50% wt. broken rice (particle wt. 0.0074 g) in between white whole long rice. The response curve for the broken rice had a smaller width and a higher maximum compared with the whole grain. In the mixture the axial dispersion of the smaller particles was less than for the larger grains, the ratio of the coefficients being approx 1.6. Of course, the shape of the larger and the smaller particles was different in this experiment. This difference in dispersion is in agreement with the qualitative observation that caked twins of coloured rice grains, being larger than the individual kernels, come out of the cylinder later. The difference in the dispersion coefficients of smaller and larger particles in a mixture implies, for an experimental set-up as described with slug feeding, that at first relatively more larger particles will be found in the response with a relatively low total concentration of marked granules, followed later by more smaller particles. There will be a tendency for the Bo numbers calculated to decrease with time. This has been found to be conilrmed in our experiments. Another variable particle property is the degree of free-flowing or cohesion between the granules. Experiments have been made with long whole rice kernels rendered sticky by moistening with syrup. With sticky particles the top of the lunar crosssectional shape of the cylinder content become broader and the kernels are taken up higher. The head over the discharge opening was greater. In one case the grains were made moderately sticky and they formed balls and agglomerates on moving in the cylinder. During an experiment with N = 11 r.p.m. the agglomerates fell apart on rolling down and hitting the bottom of the cylinder. This disintegration caused greater axial displacements of the separated kernels and the diffusion coefficient was 5 times greater than with free-flowing rice grains.
1097
R. RUTGERS
In other experiments syrup was sprayed on the rolling grains at the beginning of the cylinder, making the particles wet-sticky. Agglomerates formed were loose, but on falling to the bottom they did not fall apart into their individual grains. The whole mass showed a certain coherence and the displacements of the solid particles relative to each other certainly appeared to be restricted. The dispersion coefficients ‘calculated were made comparable with those of free-flowing dry rice by correcting the latter for differences in working content and speed of rotation with the square root formulas. M1 values found for wet-sticky rice were equal to or even somewhat lower than for dry kernels. A number of experiments were made with cut out groats, which differ in several respects from the rice grains. The average kernel weight was 0.0046 g, the particle size distribution was much wider (sieve analysis from O-15 to 0.30 cm), the shape more angular and more equidimensional, and the surface properties were also different from those of rice. As would be expected for a non-homogeneous particle size there was a trend toward lower Bodenstein numbers where these were calculated with the aid of the Brenner tables for longer times of the ascending branch of the response curve. Calculation of Bo and M1 is either less accurate or more complex. This trend was much less marked for the higher speed of rotation of 60 r.p.m. The influence of the speed of rotation on the dispersion coefficient was as in the case of rice, approx proportional to J%, but the effect of varying the working content was less clear and more experiments are needed. The cut oat groats were used with moisture contents of 7 % and 25 %. In the latter case the moisture was distributed evenly throughout the kernels by tempering for several hours. The grains were not sticky but also free-flowing. The dispersion coefficient found was 1.1 times higher than that for the dry groats. On average, the M1 value for dry cut oat groats was 1.3 times higher than for long rice. 7.
INFLUENCE OF VARYING CYLINDER DIAMETER
This important subject needs much further elucidation and only some observations can be mentioned here, based on cut oat groats in cylinders
of 76.5 cm diameter. Consistent Bodenstein numbers could be calculated for the axial mixing in these larger drums. Table 7 gives some results for the larger and for the laboratory cylinders, with the fractions of the critical speed of rotation and the Froude numbers calculated. Table
7.
Axial dispersion of cut oat groats cylinders of differing diameters
No.
Dt (cd
L N N/No (cm> (wm)
I II
76.5 76.5
230 149
15 0.31 19 0.39
O+IO97 044 0.155 0.084
E
16.0
50
60 30 0.57 0.28
0.32 0.080
C
16.0 50 30 0.28 0.080 0.18
Fr
P
0.20 0.33
Moisture content groats (%) 7 25
7
25
in
MI (cd/ SW)
0.7
1.0 0.085 0.11
0.085
As a very approximate first orientation of the diameter effect, these experiments would indicate that for a 4.8 fold increase in D,, the dispersion coefficient increases by 8.2 x or, supposing proportionality to exist with (DJDJ’, n would be only 1.34. The effect of larger diameters would be relatively small and smaller than the capacity increase with wider reactors. 8.
CONCLUSIONS
To obtain lower longitudinal dispersion, given a certain average residence time, it is advisable in the case of rotating cylinders to use low rotation speeds, high loadings, a horizontal cylinder position, no internal ribs, or rather small ones, a conical smooth entry end face and a straight outlet end face. If made roughly walled, a moderately conical outlet does not cause much more axial mixing. Surging of the particles, as compared with rolling and tumbling, gives a different mixing pattern but no greater amount of overall axial dispersion. When the reaction to be carried out involves only a reaction between solid and gas, the gas being in excess and penetrating rapidly into the interstitial voids, surging and gliding in the cylinder gives no disadvantage from the viewpoint of longitudinal dispersion. When liquid is sprayed in, however,
1098
Longitudinal
mixing of granular material flowing through a rotating cylinder-Part
good tumbling and radial mixing is required and surging must be avoided. For cohesive particles forming agglomerates and rolling down as agglomerates which burst asunder on falling on the bottom of the cylinder, axial dispersion may increase several fold. For wetsticky particles which keep their place in the rotating mass, the longitudinal dispersion coefficient may, however, be even lower as compared with dry free-flowing granules. Given a certain residence time and dispersion coefficient, the axial mixing can be reduced sharply by increasing the length of the cylinder according to the Bodenstein formula (5). For greater length, the diameter may be reduced, which lowers the M1 value and thus further increases the Bo number. Although several regularities with regard for example to the influence of the degree of filling and of the speed of rotation have been discussed in the text, it is felt that further experiments and data are needed, especially in respect of the diameter effect, before an attempt is made to give a generalized picture of longitudinal dispersion for the present type of particle flow. The problems of scaling up and down and of the dynamic similarity and of optimum dimensions of a rotating cylinder can then be handled. A Bodenstein number of 50 for flow in a reactor is generally regarded as meaning very limited axial mixing. According to Equation (4) however, the variance is then O-039 or the standard deviation 0.20. A a-figure of 0.20 would mean, of course, that 32% of the particles would have a residence time deviating by more than 20% from the mean time and 5% of the grains would deviate by more than 40%. Whether this is to be judged a high or a low variability depends on the type of reaction and the required specifications for the product. It will not always be negligible. The axial dispersion coefficient for granular material flowing through a rotating cylinder can be very low indeed and compares favourably with the coefficients mentioned in the literature for other types of flow of fluids. HOFFMANN[ 181and FRAMERS and WESTERTERP [19] have given surveys of the subject, e.g. flow of liquids and gases through tubes and annuli, flow of liquids through a rotating disc
II
reactor, fluid flow through packed and fluidized beds of solid particles. It may be said that solid particles flowing through a rotating cylinder are semi-fluidized radially by mechanical means, resulting in a rather dense phase, at least at lower speeds of rotation. Longitudinal dispersion can be made smaller than radial mixing. By choosing appropriate dimensions and operating conditions for the rotating cylinder rather large Bodenstein numbers for the process can be obtained, and this type of reactor might deserve more attention in the chemical industry [20]. Acknowledgements-Thanks are due to Mr. H. A. BOK, Managing Director of Royal Lassie Mills Ltd, for permitting publication of this work and to Mr. W. Ev~as for carefully carrying out the experiments and for his considerable further help. Discussions on the subject with Mr. W. A. BEVERLOO and, especially, with Dr. A. KLINKENBERG and Mr. S. J. VELLINGA. who allowed their method of calculation for slug-feeding to be used, helped me to clarify my present ation of the matter.
NOTATION Bodenstein number g Concentration, fractional Cylinder diameter, cm Particle diameter, cm Concentration, fractional (step-function) Working content, g Acceleration due to gravity, cm/seca Length of cylinder, cm Longitudinal dispersion coefficient, cma/sec Radial dispersion coefficient, cmz/sec Speed of rotation, r.p.m. and r.p.s., secl Critical speed of rotation, secl Number of ideal mixers in a cascade Fractional volume, degree of filling (fractional loading) Mass flow rate, g/set Radius of cylinder, cm Radial position of a particle in a cylinder, cm Reynolds number Variance of response curve Time, set Feeding time of coloured grain, set Average residence time, including &t,, set Linear velocity, cmjsec Peripheral velocity, cm/set Distance, cm Angle of internal friction Inclination of cylinder Angle of repose Porosity of collection of solid particles Relative time I/T Relative time of feeding marked particles Wall friction coefficient
R. RUTGERS 7 ? TH w
P Specific density, g/cm3 PB Bulk density, g/cm3 Standard deviation 0: Variance
Residence time, set Average residence time, set Half value time, set Angular speed, set-l
REFERENCES BRENNERH., Chem. Engng. Sci. 17 229 1962. VAN DEEMTERJ., ZIJIDERWEG F. J. and KLINKENBERG,A., Chem. Engng. Sci. 5 271 1956. KLINKENBERGA., Trans. Inst. Chem. Engrs. London, 43 T141 1965. ScrrsmrsooER A. E., Physics of Flow through Porous Media, Toronto 1960. KRAMERSH. and ALBERDAG., Chem. Ewvtg. Sci. 2 173 1953. RAOUF M. S., Continuous Mixing of solids. Diss. Wageningen 1963. MISKELLF. and MARSHALLW. R.. Chem. Enona. Proar. 52 355 1956. H~GREBEK. and LEHMANN W. S.,‘Zement KGk-Gips.‘b 133,210 1956. SMITHB. A., Trans. Am. Inst. Chem. Engrs. 38 251 1942. SAEMANW. C. and MITCHELLT. R., Chem. Engng. Progr. 50 467 1954. PAWLOWSKIJ., Chem. Zna. Tech. 26 699 1954. VIERLXNGA. and &HRE&S CH., Fiirdern und Heben 7 433, 490 1957. CREMERH. W. and DAVIEST., (Editors). Chemical Erzaineerina Practice. Volume 3. Solid Systems, London 1957. GREATHEADJ. A A. and S&&NDS W.‘IL C., Chem. &gng. >rogr. 53 i94 1957 _ SCHULZE-PILU)TG., Paper read at Achema Congress, Frankfurt 1964. RUTGERSR., Unpublished data. VANZ~~NEN D., Third Congress of European Federation of Chemical Engineers, London 1962, A54. HO~NN H., Chem. Eng&. Sci.-14 193 1961. K. R.. Elements of Chemical Reactor Desian and Ooeration. Amsterdam 1963. ‘,:i; KRAMEas H. and WESTERTERP PO1 MCCORMICKP. Y., in PERRYJ. H. (Editor). Ch.emical Engineers Handbook 20 15.-New York 1963. HI PI r31 [41 151 161 [71 PI 191 m r111 r121 1131 r141 1151 [I61 I171
R&m&-L’auteur prQente des experiences sur la dispersion axiale dans un Ccoulement continu, par exemple, de grains de riz dans un cylindre rotatif ii l’echelle du labo:atoire. 11 montre la validit de son modele d’ecoulement qui comporte la possibilite d’imposer une diffusivite longitudinale. Les coefficients de dispersion axiale ont Cte d&ermines sous diverses conditions: remplissage, vitesse de rotation, nombre de rotations, debit, inclinaison, mouvement des particules et formes de Pent& et de la sortie du cylindre. De plus, les effets de la variation de quelques proprietb des particules solides ont BtCetudies, par exemple, la forme des particules, la distribution des diverses tailles de particules, et enfIn la cohesion entre particules. Quelques experiences ont &tCfaites avec un cylindre de plus grand diametre. Les coefficients de dispersion longitudinale ont pu etre pris assaz petits, et inferieurs par exemple a 0,04 cm2/sec. tres inferieurs done a ceux obtenus avec la plupart des autres types d’appareils avec ecoulement de fluides. Zusarnmenfasstmg-Es werden Versuche beschrieben, bei denen langkiirniger Reis kontinuierlich durch einen rotierenden Experimentier-Zylinder Ilie&, urn dessen axiale Verteilung zu untersuchen. Eine Modellvorstellung, bei der die Pfropfenstriimung von axialer Riickvermischung tiberlagert wird, erwies sich als gtiltig. Axiale Mischkoefbzienten wurden unter verschiedenen Bedingungen bestimmt, wobei der Durchsatz, die Rotationsgeschwindigkeit, die lineare Stromungsgeschwindigkeit, die Zylinderneigung, die Art der transportierten Partikel und die Gestaltung der zylindrischen Eintrittsbzw. Austrittsoffnungen verandert wurden. Dariiber hinaus wurden einige Partikeleigenschaften verandert, wie z.B. ihre Form, ihre KorngriiBenverteilung und ihre gegenseitige Haftung. Auch von einigen Experimenten bei vergriigertem Zylinderdurchmesser wird berichtet. Die axialen Mischkoeffizienten kijnnen sehr klein sein (herab bis 0,04 cma/sec) und liegen damit niedriger als in den meisten Stromungsprozessen und Reaktoren.
1100