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A further contribution to the evaluation of the Jenike method for design of mass flow hoppers
Powder TechnoZogy, 10 (1974) 51-58 @ Elsevier Sequoia S-A., Lausanne -Printed
A Further
Contributionto
R.K. ECKHOFF Powder Bergen
Research (Norway)
(Received
in The Netherlands
the Evaluation
for Design of Mass Flow Hoppers
of the Jenike Method
and P.-G. LEVERSEN Laboratory,
February
23,1973;
Chr. Michelsen
Institute,
in revised form January
Dept.
of Applied
Physics,
Nygaardsgaten
114. N-5000
17, 1974)
-
SUMMARY
A critical comparison has been carried out between values for the minimum outlet slot width and the minimum hopper wall slope for mass flow predicted by the Jenike method, and values obtained from experiments on a full-scale silo with symmetrical wedge-shaped hopper. The data for the minimum slot width were obtained for a fine Sic powder whilst a coarser, free-flowing powder of the same material was used to determine the critical slope
of the hopper walls. It was found that the denike method overdesigned the critical hopp3r slope by 8 - 10”. It was further found that the slot width was overdesigned by from 0 to over 100% depending on the way in which the failure loci were extrapolated into the range of small normal stresses.
1. INTRODUCI’ION The work of Jenike [l] has led to a method for the design of storage hoppers for particulate solids in which stable powder arches cannot form across the outlet. The theory and experimental techniques have been widely used in the design of hoppers and have given rise to extensive research in this field. In spite of this, the literature gives very little information on the correlation between the critical hopper design predicted by the Jenike method and the corresponding parameters for full-scale hoppers. Walker. [Z] concluded that the Jcnike method overdesigns the hopper outlet quite considerably. More recently Wright [3] published an interesting
series of results extracted from his thesis from the University of Bradford 143 _ In this case it was shown that for iron ores the wall slope required by the Jenike method for mass flow was overdesigned by 5 - lo”, whereas the predicted critical outlet size agreed closely with the experimental result. It thus appears that the published data are contradictory and that there is a need for further work on this problem_ The work presented in this paper was planned and carried out to obtain further information on the relation between predicted and experimental results. 2. PROPERTIES
OF THE
POWDERS
USED
The bulk density, moisture content, and mean particle size for the two powders used are given in Table 1. The failure properties of the two powders were determined by means of a standard Jenike “Flow Factor Tester”. The standard procedure recommended by Jenike [1] for preparing the powder specimens in the shear cell was employed. The details of the experimental procedure used and the method of obtaining points on the failure loci from raw data etc. are given elsewhere [5] _ Preliminary experiments indicated that the coarse powder was almost free flowing, as shown by the family of three failure loci given in Fig. 1. As can be seen, the cohesion is negligible for all three states of consolidation. The fine powder was, however, cohesive. Six failure loci, referring to six different states of consolidation, were determined, each point on a locus being based on six single shear cell
52 TA%LE
1 T
Powder
Bulk density (g/cm )
Moisture content W)
Approx. mean particle size (Ctm)
Sic, coarse SIC, fine
1.3 0.8*
0.01 0.10
20 1
* In the region of the outlet
(hulk density
kp/crt?
0.02 -
of arch) _ 0.01 -
tests. The family of failure loci for the fine powder is given in Fig. 2. To determine the effect of time consolidation for the fine powder, a 100 hours time test was carried out at two different states of consolidation. For each consolidation state six points on the failure locus were determined. The two loci are given in Fig. 3 together with the, two corresponding loci from Fig. 2 for no time consolidation.
3. DETERMINATION FROM
FAILURE
OF FAILURE
0.02
Cl01
Cl03 kp/cm2
cr
Fig. 2. Family
of six failure loci for fine Sic powder.
FUNCTIONS
LOCI DATA
The failure functions for the two powders are given in Fig. 4. As can be seen, the failure function for the coarse powder practically coincides with the horizontal axis, as would be expected from Fig_ 1. In the case of the fine powder, however, the picture is more complicated. As shown in Fig. 4, two alternatives for the failure function are indicated. The reason for this emerges I
0.02
U
LDOI
of
three
failure
loci
Fig. 3. Effect of time consolidation for the fine Sic powder.
on the failure loci
Fig. 4. used.
two
om*
1810
a
Fig. 1. Family powder.
kplcd
for
wrd
coarse
Sic
Failure
functions
for
the
Sic
powders
53
T kp/cm’
0.0
1
0.02
0.03
0.0‘
0
kp/cm’
Fig. 5. Illustration of the two alternative methods extrapolating the failure loci.
for
from a closer consideration of Fig. 2, from which it can be observed that all the six failure loci exhibit a characteristic fall-off in shear strength in the range of normal stresses below 0.005 - 0.01 kp/cm2, which is greater than the fall-off to be expected from the trend exhibited for higher normal stresses. Hence, two extreme possibilities for extrapolating the failure locus into the range of smaller normal stresses exist. This is illustrated in detail in Fig. 5 by means of one of the failure loci given in Fig. 2. It is seen that the two methods of extrapolation produce two different values of the unconfined failure stress, f, , and this is the explanation for the two alternative failure functions for the fine powder shown in Fig. 4. It has been observed both in the present investigation and in earlier work carried out at CM1 that when the normal stress was decreased below the critical range mentioned above, the lid of the shear cell tended to rise slightly from the powder surface opposite the side of the bracket transmitting the shear force. It should be noted that this phenomenon was also observed by Walker [Z] _ As can be deduced from Fig. 6, the lifting of the lid may possibly be traced back to the fact that the shear force is applied to the bracket below
Normal
for
c c
1
NLW
pinJ
1 I
Fig. 6. Locations of pins transmitting the ring of the shear cell.
I
the shear force to
I
I
I
I
0.01
0.0
U
Fig. 7. Failure bracket pin.
loci
obtained
with
2
kp/cm2
original
and new
the position of the pin which transmits the force from the bracket to the ring of the cell, and thus imposes a tilting moment upon the lid. The question was now whether this tilting effect, which was suspected of inducing an uneven normal stress distribution in the cell, was the origin of the marked fall-off of the shear strength of the powder specimen in the cell in the range of low normal stresses. To investigate this, the bracket was fitted with a new pin as indicated in Fig. 6, so that the tilting moment with respect to the lid was reduced considerably. The effect of this modification was, however, negligible. Figure ‘7 shows an example of two loci, one obtained with the original and one with the new pin, other conditions being identical. As can be seen, the two loci demonstrate good agreement on the essential point, namely the fall-off of the shear strength in the range of small normal stresses. During the experiments with the new pin the cell was observed very carefully in the course of the shearing process, and it was noticed that, with normal stresses in the critical range under discussion, the plane surface of the powder in the cell gradually appeared to attain the shape of a convex upwards lens, the contact surface between powder and lid thus being reduced to a smaller area than the area of the cell. This observation, as well as the fact that failure loci having sharp bends of the type
54 4. PREDICTION PARHMETERS
0.005
0.010
0.020
0.015
U
kplcmZ
Fig. 8. “Wall yield loci” for the two powders in combination with the wall material of the test silo.
in Fig. 2 seem improbable, suggest that the experimental points in the range of very low normal stresses should be treated with caution, since the stress distribution in the cell is under such circumstances likely to be uneven. What is not known, however, is the extent to which this has any significant effect on the measured shear stresses at failure. Hence, one is left with the two extreme possibilities illustrated in Fig. 4 for performing the extrapolation of the failure loci into the range of low normal stresses, the one pass sibility being based on the rejection of the points below 0.005 - 0.01 experimental the other on the acceptance of these kp/cm2, points. The actual course of the failure locus is most likely to be located somewhere between the two extremes, but its actual position is not known. shown
OF CRITICAL
HOPPER
Mild steel was selected as the wall material for the full-scale silo and in Fig. 8 the wall yield loci for the two powders in combination with this material are given. Each experimental point in Fig. 8 is based on six individual experiments. By applying the design procedure for symmetrical, plane mass flow hoppers given by Jenike [l] to the data given in Figs. 4 and 8, the critical parameters for the slot width and the angle between the inclined hopper wall and the vertical were predicted. The critical parameters are given in Table 2. As can be seen from Fig. 2, most of the failure loci consist of two approximately linear parts. On the basis of this situation, a computer programme based on linear leastsquare regression analysis was made for calculating f, and o1 from the experimental shear cell data [S] . The i intervals given in Table 2 for the critical slot widths are the 97.5% confidence interval obtained from the statistical computer analysis. Due to lack of data, such confidence limits could not be obtained for the critical outlet widths corresponding to 100 hours time consolidation. The predicted Alt. 1 critical slot width is the result obtained when using the Alt. 1 failure function. The predicted Alt. 2 critical slot widths are the result obtained by: (a) using as the failure function the straight line through the two lowest Alt. 2 points, or (b) using instead the least-square regression line through
TABLE 2 Predicted critical hopper parameters Powder
Sic, coarse
Sic, fine
Time consolation
Predicted critical hopp;r wall angle, en (deg)
Predicted Alt. 1 critical slot width, B (-1
Predicted AR. 2 critical slot width, B (cm)
None
15.5
=0
=0
63 hours
-
=0
=0
None
16
14.4 f 0.4
100 hours
-
21.5
(a) 6.2 (b) 9.2 * 11.0 14.2
(a) Based on the two lowest points (b) Based on all points.
55
all the Ah. 2 points as the failure function. Due to the scatter of the points around this line, however, the confidence interval on the outlet width obtained in this way will be quite wide, e.g. f 11.0 cm at the 97.5% level. It should be emphasized that the predicted hopper wall angles refer to the recommended dotted lines in the design graphs in Jenike’s Bull. 123 [l] _
5. FULL-SCALE
SILO EXPERIMENTS
5.1 The experimental silo The silo, which was made of simple wood structures, is illustrated in Fig. 9. The inclined walls of the symmetrical plane flow hopper were adjustable both with respect to the outlet slot width and the wall angles. All the internal surfaces of the silo were lined with l-mm steel plate of the same quality as the sample used for establishing the wall yield loci in Fig. 8. The hopper wall angle could be varied between approximately 0 and 90’ and the outlet width from 0 to 50 cm. Sufficient headroom was available below the hopper to permit the powder flowing out of the silo to be collected in a box which could afterwards be lifted to the silo top and its content refilled into the silo. When discharging the silo content into t.he box, provision was made for continuous monitoring of the weight of the box.
5.2 Critical hopper wail slopes for mass flow The purpose of this series of experiments was to test the validity of the critical hopper wall slope predicted by the Jenike method for the coarse powder. In order to obtain information with respect to the flow pattern of the powder in the silo during discharge, a number of 7 by 1 cm labelled strips of aluminium plate were initially placed carefully above each other at known distances in the core of the powder bed, as the silo was filled up. During the discharge process a net capable of catching the aluminium strips was placed just below the outlet, and as soon as a strip was observed the weight of the powder accumulated in the box below was recorded instantaneously_ Since the powder in question was practically incompressible, the accumulated weight was proportional to the accumulated bulk volume of powder. The results from the experiments with the coarse powder are given in Fig. 10, where the accumulated weight has been plotted as a function of the initial positions of the strips in the silo, in terms of heights above the silo exit. The dotted curves are those which would have been expected for idealized mass flow in which horizontal planes in the powder would remain horizontal right down to the silo exit. As is well known, however, this flow pattern does not occur in practice. Whilst the horizontal plane powder elements in a mass flow silo will remain largely undistorted dur-
Weight kg
100
11
Fig. 9. Illustration of the 1 m3 capacity in the experiments.
test silo used
8~5
cm,
G$=
23.5’
Fig. 10. Results from investigations of flow patterns of the coarse Sic powder in the test silo.
56
ing flow downwards in the parallel part of the silo, their shape will be gradually distorted as the!y proceed through the converging hopper part. In the present context this means that an aluminium strip positioned in the core of the silo will always leave the silo exit before all the powder initially associated with it has been discharged. This in turn means that in real mass flow, the accumulated weight curve will be positioned somewhat to the right of the curve for the idealized flow pattern. On this background it is interesting to observe that the curve in Fig. 10 for 0; = 23.5” appears to exhibit the trend to be expected from real mass flow. Mass flow was also indicated by the fact that when discharging the silo, the powder surface, when observed from above, remained smooth and horizontal until it approached the transition region between the parallel part and the converging hopper. At that point a funnel was gradually formed. Figure 10 also gives the curve obtained with 8 $ = 25”, which is different from the 23.5” curve in two respects. In the initial phase it appears to coincide with the theoretical mass flow curve, but after the discharge of the label initially positioned about 20 cm abo le the exit, a linear trend is obtained. The slope of this straight line corresponds to a thickl:ers of the moving core of about 17 cm. When the label initially placed about 70 cm above the exit has just escaped the siio, the thickness of the moving powder core increases fairly suddenly to roughly 26 cm. This corresponds approximately to the point where the powder level in the silo approaches the transition to the converging hopper. It is thus seen that the rather moderate increase of 0: from 23.5” to 25’ produces a marked change of the flow pattern from mass flow to a transition pattern with distinct funnel flow features. It should be noted that whilst the 23.5” experiments were run with an outlet width of 5 cm, the width was 2 cm for 0 k equal to 25”. This could be a -eason why the 25” curve falls above the 23.5” curve during the first part of the discharge process. A comprehensive interpretation of this rather unexpected effect has not yet, however, been found. 5.3 Critical outlet width Initial experiments revealed lsowder was partly fluidized
that the fine when being
poured carefully from the storage box into the silo. This effect was clearly undesirable, since critical outlets obtained with partly fluidized powder would not under any circumstances be comparable with those predicted by the Jenike method. After having abandoned the possibility of finding a filling technique which did not cause the powder to absorb air, it was decided to leave the silo after filling, until the air introduced during filling had escaped. A direct method for assessing the correct time required for adequate deaeration was, however, not available, and hence the following two alternative procedures were employed to overcome this problem. In the first method, the hopper slot width was initially adjusted. to a value less than the expected critical width. The silo was then filled and the powder allowed to deaerate for three hours, during which the powder level in the silo stabilized and the powder redeveloped its original “dead” appearance. The possibility existed that the powder in the course of this period had also been subjected to a significant time consolidation. In order to determine the critical slot width, however, the hopper walls, after opening the outlet, were adjusted stepwise to increase the outlet width by 0.4 - 1.0 cm per step, and during this process a possibie time consolidation effect was likely to disappear. The critical slot width was then takea as the width at which the arch was broken and the flow of powder started. This experiment was then repeated five times to give six readings for the critical opening. In the second method used, the opening was set in a fixed position, closed and the silo filled with powder. After about half an hour, which was the time required for the powder surface in the silo to reach an approximately steady ievel, the outlet was opened. In case of flow, the experiment was repeated with a smaller opening, until a width was reached at which a:1 arch was formed. In this case a moderate vibration was introduced in the powder by knocking at the hopper wall, and if flow was then induced, it was interrupted for intervals of 10 - 60 seconds to test whether stable arches would then form. The smallest slot width for which an arch did not form after such interruptions of flow was taken as the critical value. The results from the experiments discussed
57 TABLE 3 Data from method 1 for determining the critical outlet width
TABLE 5 Real critical outlet width after 100 hours time consolidation
Initial width (cm1
Number of StePS
Critical width (cm)
Outlet width (cm1
Comments
4.0 5.8 4.8 3.7 4.3 2.4
1 2 3 1* 1* 3
6.5 6.5 6.4 4.2 5.5 3.1
15
Steady flow started immediately after opening of outlet Steady flow started a few seconds after opening of outlet No flow after opening of outlet. Flow was initiated by knocking at the hopper wall and thus destroying the time consolidation effect
13 10
* Rather violent adjustment of hopper walls.
above are given in Tables 3 and 4. As can be seen, both methods yield critical slot widths for the powder tested of about 5 +l cm, which is less than half the value predicted by the Jenike procedure, when using the Alt. 1 failure function, but fairly close to the result obtained when using the two lowest Alt. 2 points. In order to test the predicted critical slot width corresponding to 100 hours time consolidation, method 2 described above was used. The results are shown in Table 5, from which it can be seen that the critical width is somewhere in the range of 10 - 13 cm. Again, this is only about half the value of 21.5 cm predicted by the Jenike method when using the Alt. 1 failure function. It should finally be noted that mass flow appeared to be present in the test with the fine powder, since the powder surface in the silo, when observed from above, remained
TABLE 4 Data from method 2 for determining the critical outlet width Outlet width
Comments
9.6 8.1 6.2
Flow initiated immediately the outlet
4.3 4.2
Flow initiated after knocking at hopper walls, but in s-jme cases stable arches were formed after interruptions of flow
3.2
Erratic unstable flow induced by knocking at the hopper walls. Stable arches formed after short interrup tions of flow.
by opening
smooth and horizontal until it reached transition to the hopper part.
the
CONCLUSION
As far as prediction of the critical !iopper wall slope for mass flow is concerns& the results obtained in the present investigation agree well with the results obtained by Wright [3], since in both cases an overdesign of the order of 8 - 10” was observed, and this indicates that the recommended, dotted lines in Jenike’s design graphs [l] involve this order of overdesign. For practical design purposes, however, where the possibility of uncontrolled variations in powder properties and in the wall friction must be accounted for, it might be wise, as suggested by Wright, to maintain the procedure recommended by Jenike. When it comes to the prediction of the critical outlet width for no-arching, the situation turned out to be more complicated, mainly due to difficulties in deciding how the failure loci should be extrapolated into the low normal stress region. Hence, a corresponding uncertainty as to the ;nagnitude of the unconfined failure stress, f, , which is derived on the basis of the extrapolated part of a failure locus, was the result. With the particular powder used, a fine quality Sic, this problem appeared to be particullarly exaggerated, since mosi UPthe failure loci appeared to consist of two approximately linear parts with diffzrent slopes, the slope of the low normal stress part being higher than that of the main part. Hence, two distinct alternatives for extrapolating the failure !oci were available, yielding f, values which differed by a factor of approximately two.
58
Although some experimental evidence relating the increase in the slopes of the failure loci in the low normal stress range to inadequate stress distribution in the shear cell during shear was produced, a quantitative assessment of the effect of this on the measured shear stresses at failure was not possible. If, however, the low normal stress points were totally neglected, the remaining points for most of the failure loci fell on straight lines with high degrees of precision, indicating that a linear extrapolation on the basis of these points was a sensible approach_ In the present case this approach produced an overdesign on the critical outlet width in excess of 100%. If, on the other hand, the extrapolation was based on the low normal stress point, a critical outlet width of approximately the same magnitude as the value obtained in the full-scale silo experiments was obtained. In this case, however, an adequate explanation for the distinct increase in the slope of the failure loci in the range of low normal stresses remains to be found. It should be pointed out that another, very cohesive powder being at present subjected to careful shear cell tests in our laboratory does not exhibit the very sharp transition between two parts of the failure loci as demonstrated by the Sic powder, the increase in slope occurring more gradually over a fairly wide range of normal stresses. Nevertheless, it appears that even the failure loci for these powders present considerable difficulties when it comes to the extrapolation into the region of low normal stresses. We feel that this problem of performing an adequate extrapolation of the failure loci represents an inherent difficulty in the Jenike method for design of mass flow hoppers. This problem can probably be considerably reduced by combining the shear cell tests with measurements for tensile strength, and perhaps also cohesion. In principle, a discrepancy between predicted and real critical outlet widths can arise either from an inadequate method of prcdiction, or from a lack of correspondence between the conditions for which the theory is
developed and those actually prevailing in the test silo. It is the authors’ opinion that in the present case, the full-scale silo experiments were carried out under conditions corresponding closely to the dynamic conditions presumed in the theory. Hence, it is felt that the discrepancy observed must be attributed to shortcomings in the method of prediction_ This in turn can be due either to inadequate assumptions in the theory or to shortcomings in the experimental technique used for determining the failure function, or to a combination of both factors. At present the picture is far from clear, and further systematic research aimed at resolving the problems should be encouraged. ACKNOWLEDGEMENTS
The full-scale silo was made by and the experiments on it carried out at Norton A/S, Lillesand, Norway, and the authors wish to express their gratitude to Mr. B. Aune for invaluable help. Thanks are due also to the Royal Korwegian Council for Scientific and Industrial Research for the financial support without which this investigation would not have been possible. Finally, the authors are indebted to director Dr. J.A. Andersen, CMI, for stimulating encouragement.
REFERENCES A.W. Jenike, Storageand fl-,w of solids, Bull. 123, Eng. Exp. Sta., Utah State Univ., 1964. 2 D.M. Walker, A basis for bunker design, Powder Technol., 1 (1967) 228. H. Wright, An evaluation of the Jenike bunker design method, Trans. ASME, Paper No. 72-MH-7, 1972. H. Wright, Bunker design for iron ores, Ph.D. Thesis, Univ. of Bradford, 1970. P-G. Leversen, An investigation of the flow properties of powders in silos, Thesis, Tech. Univ. of Norway, Trondheim, 1972 (in Norwegian). L. Mtirkrid, Estimation of fc and ~1 from shear cell da’s by means of regression analysis, assuming linear failure loci, CMI-Report, 1972 (in Norwegian). I
A Further
Contributionto
R.K. ECKHOFF Powder Bergen
Research (Norway)
(Received
in The Netherlands
the Evaluation
for Design of Mass Flow Hoppers
of the Jenike Method
and P.-G. LEVERSEN Laboratory,
February
23,1973;
Chr. Michelsen
Institute,
in revised form January
Dept.
of Applied
Physics,
Nygaardsgaten
114. N-5000
17, 1974)
-
SUMMARY
A critical comparison has been carried out between values for the minimum outlet slot width and the minimum hopper wall slope for mass flow predicted by the Jenike method, and values obtained from experiments on a full-scale silo with symmetrical wedge-shaped hopper. The data for the minimum slot width were obtained for a fine Sic powder whilst a coarser, free-flowing powder of the same material was used to determine the critical slope
of the hopper walls. It was found that the denike method overdesigned the critical hopp3r slope by 8 - 10”. It was further found that the slot width was overdesigned by from 0 to over 100% depending on the way in which the failure loci were extrapolated into the range of small normal stresses.
1. INTRODUCI’ION The work of Jenike [l] has led to a method for the design of storage hoppers for particulate solids in which stable powder arches cannot form across the outlet. The theory and experimental techniques have been widely used in the design of hoppers and have given rise to extensive research in this field. In spite of this, the literature gives very little information on the correlation between the critical hopper design predicted by the Jenike method and the corresponding parameters for full-scale hoppers. Walker. [Z] concluded that the Jcnike method overdesigns the hopper outlet quite considerably. More recently Wright [3] published an interesting
series of results extracted from his thesis from the University of Bradford 143 _ In this case it was shown that for iron ores the wall slope required by the Jenike method for mass flow was overdesigned by 5 - lo”, whereas the predicted critical outlet size agreed closely with the experimental result. It thus appears that the published data are contradictory and that there is a need for further work on this problem_ The work presented in this paper was planned and carried out to obtain further information on the relation between predicted and experimental results. 2. PROPERTIES
OF THE
POWDERS
USED
The bulk density, moisture content, and mean particle size for the two powders used are given in Table 1. The failure properties of the two powders were determined by means of a standard Jenike “Flow Factor Tester”. The standard procedure recommended by Jenike [1] for preparing the powder specimens in the shear cell was employed. The details of the experimental procedure used and the method of obtaining points on the failure loci from raw data etc. are given elsewhere [5] _ Preliminary experiments indicated that the coarse powder was almost free flowing, as shown by the family of three failure loci given in Fig. 1. As can be seen, the cohesion is negligible for all three states of consolidation. The fine powder was, however, cohesive. Six failure loci, referring to six different states of consolidation, were determined, each point on a locus being based on six single shear cell
52 TA%LE
1 T
Powder
Bulk density (g/cm )
Moisture content W)
Approx. mean particle size (Ctm)
Sic, coarse SIC, fine
1.3 0.8*
0.01 0.10
20 1
* In the region of the outlet
(hulk density
kp/crt?
0.02 -
of arch) _ 0.01 -
tests. The family of failure loci for the fine powder is given in Fig. 2. To determine the effect of time consolidation for the fine powder, a 100 hours time test was carried out at two different states of consolidation. For each consolidation state six points on the failure locus were determined. The two loci are given in Fig. 3 together with the, two corresponding loci from Fig. 2 for no time consolidation.
3. DETERMINATION FROM
FAILURE
OF FAILURE
0.02
Cl01
Cl03 kp/cm2
cr
Fig. 2. Family
of six failure loci for fine Sic powder.
FUNCTIONS
LOCI DATA
The failure functions for the two powders are given in Fig. 4. As can be seen, the failure function for the coarse powder practically coincides with the horizontal axis, as would be expected from Fig_ 1. In the case of the fine powder, however, the picture is more complicated. As shown in Fig. 4, two alternatives for the failure function are indicated. The reason for this emerges I
0.02
U
LDOI
of
three
failure
loci
Fig. 3. Effect of time consolidation for the fine Sic powder.
on the failure loci
Fig. 4. used.
two
om*
1810
a
Fig. 1. Family powder.
kplcd
for
wrd
coarse
Sic
Failure
functions
for
the
Sic
powders
53
T kp/cm’
0.0
1
0.02
0.03
0.0‘
0
kp/cm’
Fig. 5. Illustration of the two alternative methods extrapolating the failure loci.
for
from a closer consideration of Fig. 2, from which it can be observed that all the six failure loci exhibit a characteristic fall-off in shear strength in the range of normal stresses below 0.005 - 0.01 kp/cm2, which is greater than the fall-off to be expected from the trend exhibited for higher normal stresses. Hence, two extreme possibilities for extrapolating the failure locus into the range of smaller normal stresses exist. This is illustrated in detail in Fig. 5 by means of one of the failure loci given in Fig. 2. It is seen that the two methods of extrapolation produce two different values of the unconfined failure stress, f, , and this is the explanation for the two alternative failure functions for the fine powder shown in Fig. 4. It has been observed both in the present investigation and in earlier work carried out at CM1 that when the normal stress was decreased below the critical range mentioned above, the lid of the shear cell tended to rise slightly from the powder surface opposite the side of the bracket transmitting the shear force. It should be noted that this phenomenon was also observed by Walker [Z] _ As can be deduced from Fig. 6, the lifting of the lid may possibly be traced back to the fact that the shear force is applied to the bracket below
Normal
for
c c
1
NLW
pinJ
1 I
Fig. 6. Locations of pins transmitting the ring of the shear cell.
I
the shear force to
I
I
I
I
0.01
0.0
U
Fig. 7. Failure bracket pin.
loci
obtained
with
2
kp/cm2
original
and new
the position of the pin which transmits the force from the bracket to the ring of the cell, and thus imposes a tilting moment upon the lid. The question was now whether this tilting effect, which was suspected of inducing an uneven normal stress distribution in the cell, was the origin of the marked fall-off of the shear strength of the powder specimen in the cell in the range of low normal stresses. To investigate this, the bracket was fitted with a new pin as indicated in Fig. 6, so that the tilting moment with respect to the lid was reduced considerably. The effect of this modification was, however, negligible. Figure ‘7 shows an example of two loci, one obtained with the original and one with the new pin, other conditions being identical. As can be seen, the two loci demonstrate good agreement on the essential point, namely the fall-off of the shear strength in the range of small normal stresses. During the experiments with the new pin the cell was observed very carefully in the course of the shearing process, and it was noticed that, with normal stresses in the critical range under discussion, the plane surface of the powder in the cell gradually appeared to attain the shape of a convex upwards lens, the contact surface between powder and lid thus being reduced to a smaller area than the area of the cell. This observation, as well as the fact that failure loci having sharp bends of the type
54 4. PREDICTION PARHMETERS
0.005
0.010
0.020
0.015
U
kplcmZ
Fig. 8. “Wall yield loci” for the two powders in combination with the wall material of the test silo.
in Fig. 2 seem improbable, suggest that the experimental points in the range of very low normal stresses should be treated with caution, since the stress distribution in the cell is under such circumstances likely to be uneven. What is not known, however, is the extent to which this has any significant effect on the measured shear stresses at failure. Hence, one is left with the two extreme possibilities illustrated in Fig. 4 for performing the extrapolation of the failure loci into the range of low normal stresses, the one pass sibility being based on the rejection of the points below 0.005 - 0.01 experimental the other on the acceptance of these kp/cm2, points. The actual course of the failure locus is most likely to be located somewhere between the two extremes, but its actual position is not known. shown
OF CRITICAL
HOPPER
Mild steel was selected as the wall material for the full-scale silo and in Fig. 8 the wall yield loci for the two powders in combination with this material are given. Each experimental point in Fig. 8 is based on six individual experiments. By applying the design procedure for symmetrical, plane mass flow hoppers given by Jenike [l] to the data given in Figs. 4 and 8, the critical parameters for the slot width and the angle between the inclined hopper wall and the vertical were predicted. The critical parameters are given in Table 2. As can be seen from Fig. 2, most of the failure loci consist of two approximately linear parts. On the basis of this situation, a computer programme based on linear leastsquare regression analysis was made for calculating f, and o1 from the experimental shear cell data [S] . The i intervals given in Table 2 for the critical slot widths are the 97.5% confidence interval obtained from the statistical computer analysis. Due to lack of data, such confidence limits could not be obtained for the critical outlet widths corresponding to 100 hours time consolidation. The predicted Alt. 1 critical slot width is the result obtained when using the Alt. 1 failure function. The predicted Alt. 2 critical slot widths are the result obtained by: (a) using as the failure function the straight line through the two lowest Alt. 2 points, or (b) using instead the least-square regression line through
TABLE 2 Predicted critical hopper parameters Powder
Sic, coarse
Sic, fine
Time consolation
Predicted critical hopp;r wall angle, en (deg)
Predicted Alt. 1 critical slot width, B (-1
Predicted AR. 2 critical slot width, B (cm)
None
15.5
=0
=0
63 hours
-
=0
=0
None
16
14.4 f 0.4
100 hours
-
21.5
(a) 6.2 (b) 9.2 * 11.0 14.2
(a) Based on the two lowest points (b) Based on all points.
55
all the Ah. 2 points as the failure function. Due to the scatter of the points around this line, however, the confidence interval on the outlet width obtained in this way will be quite wide, e.g. f 11.0 cm at the 97.5% level. It should be emphasized that the predicted hopper wall angles refer to the recommended dotted lines in the design graphs in Jenike’s Bull. 123 [l] _
5. FULL-SCALE
SILO EXPERIMENTS
5.1 The experimental silo The silo, which was made of simple wood structures, is illustrated in Fig. 9. The inclined walls of the symmetrical plane flow hopper were adjustable both with respect to the outlet slot width and the wall angles. All the internal surfaces of the silo were lined with l-mm steel plate of the same quality as the sample used for establishing the wall yield loci in Fig. 8. The hopper wall angle could be varied between approximately 0 and 90’ and the outlet width from 0 to 50 cm. Sufficient headroom was available below the hopper to permit the powder flowing out of the silo to be collected in a box which could afterwards be lifted to the silo top and its content refilled into the silo. When discharging the silo content into t.he box, provision was made for continuous monitoring of the weight of the box.
5.2 Critical hopper wail slopes for mass flow The purpose of this series of experiments was to test the validity of the critical hopper wall slope predicted by the Jenike method for the coarse powder. In order to obtain information with respect to the flow pattern of the powder in the silo during discharge, a number of 7 by 1 cm labelled strips of aluminium plate were initially placed carefully above each other at known distances in the core of the powder bed, as the silo was filled up. During the discharge process a net capable of catching the aluminium strips was placed just below the outlet, and as soon as a strip was observed the weight of the powder accumulated in the box below was recorded instantaneously_ Since the powder in question was practically incompressible, the accumulated weight was proportional to the accumulated bulk volume of powder. The results from the experiments with the coarse powder are given in Fig. 10, where the accumulated weight has been plotted as a function of the initial positions of the strips in the silo, in terms of heights above the silo exit. The dotted curves are those which would have been expected for idealized mass flow in which horizontal planes in the powder would remain horizontal right down to the silo exit. As is well known, however, this flow pattern does not occur in practice. Whilst the horizontal plane powder elements in a mass flow silo will remain largely undistorted dur-
Weight kg
100
11
Fig. 9. Illustration of the 1 m3 capacity in the experiments.
test silo used
8~5
cm,
G$=
23.5’
Fig. 10. Results from investigations of flow patterns of the coarse Sic powder in the test silo.
56
ing flow downwards in the parallel part of the silo, their shape will be gradually distorted as the!y proceed through the converging hopper part. In the present context this means that an aluminium strip positioned in the core of the silo will always leave the silo exit before all the powder initially associated with it has been discharged. This in turn means that in real mass flow, the accumulated weight curve will be positioned somewhat to the right of the curve for the idealized flow pattern. On this background it is interesting to observe that the curve in Fig. 10 for 0; = 23.5” appears to exhibit the trend to be expected from real mass flow. Mass flow was also indicated by the fact that when discharging the silo, the powder surface, when observed from above, remained smooth and horizontal until it approached the transition region between the parallel part and the converging hopper. At that point a funnel was gradually formed. Figure 10 also gives the curve obtained with 8 $ = 25”, which is different from the 23.5” curve in two respects. In the initial phase it appears to coincide with the theoretical mass flow curve, but after the discharge of the label initially positioned about 20 cm abo le the exit, a linear trend is obtained. The slope of this straight line corresponds to a thickl:ers of the moving core of about 17 cm. When the label initially placed about 70 cm above the exit has just escaped the siio, the thickness of the moving powder core increases fairly suddenly to roughly 26 cm. This corresponds approximately to the point where the powder level in the silo approaches the transition to the converging hopper. It is thus seen that the rather moderate increase of 0: from 23.5” to 25’ produces a marked change of the flow pattern from mass flow to a transition pattern with distinct funnel flow features. It should be noted that whilst the 23.5” experiments were run with an outlet width of 5 cm, the width was 2 cm for 0 k equal to 25”. This could be a -eason why the 25” curve falls above the 23.5” curve during the first part of the discharge process. A comprehensive interpretation of this rather unexpected effect has not yet, however, been found. 5.3 Critical outlet width Initial experiments revealed lsowder was partly fluidized
that the fine when being
poured carefully from the storage box into the silo. This effect was clearly undesirable, since critical outlets obtained with partly fluidized powder would not under any circumstances be comparable with those predicted by the Jenike method. After having abandoned the possibility of finding a filling technique which did not cause the powder to absorb air, it was decided to leave the silo after filling, until the air introduced during filling had escaped. A direct method for assessing the correct time required for adequate deaeration was, however, not available, and hence the following two alternative procedures were employed to overcome this problem. In the first method, the hopper slot width was initially adjusted. to a value less than the expected critical width. The silo was then filled and the powder allowed to deaerate for three hours, during which the powder level in the silo stabilized and the powder redeveloped its original “dead” appearance. The possibility existed that the powder in the course of this period had also been subjected to a significant time consolidation. In order to determine the critical slot width, however, the hopper walls, after opening the outlet, were adjusted stepwise to increase the outlet width by 0.4 - 1.0 cm per step, and during this process a possibie time consolidation effect was likely to disappear. The critical slot width was then takea as the width at which the arch was broken and the flow of powder started. This experiment was then repeated five times to give six readings for the critical opening. In the second method used, the opening was set in a fixed position, closed and the silo filled with powder. After about half an hour, which was the time required for the powder surface in the silo to reach an approximately steady ievel, the outlet was opened. In case of flow, the experiment was repeated with a smaller opening, until a width was reached at which a:1 arch was formed. In this case a moderate vibration was introduced in the powder by knocking at the hopper wall, and if flow was then induced, it was interrupted for intervals of 10 - 60 seconds to test whether stable arches would then form. The smallest slot width for which an arch did not form after such interruptions of flow was taken as the critical value. The results from the experiments discussed
57 TABLE 3 Data from method 1 for determining the critical outlet width
TABLE 5 Real critical outlet width after 100 hours time consolidation
Initial width (cm1
Number of StePS
Critical width (cm)
Outlet width (cm1
Comments
4.0 5.8 4.8 3.7 4.3 2.4
1 2 3 1* 1* 3
6.5 6.5 6.4 4.2 5.5 3.1
15
Steady flow started immediately after opening of outlet Steady flow started a few seconds after opening of outlet No flow after opening of outlet. Flow was initiated by knocking at the hopper wall and thus destroying the time consolidation effect
13 10
* Rather violent adjustment of hopper walls.
above are given in Tables 3 and 4. As can be seen, both methods yield critical slot widths for the powder tested of about 5 +l cm, which is less than half the value predicted by the Jenike procedure, when using the Alt. 1 failure function, but fairly close to the result obtained when using the two lowest Alt. 2 points. In order to test the predicted critical slot width corresponding to 100 hours time consolidation, method 2 described above was used. The results are shown in Table 5, from which it can be seen that the critical width is somewhere in the range of 10 - 13 cm. Again, this is only about half the value of 21.5 cm predicted by the Jenike method when using the Alt. 1 failure function. It should finally be noted that mass flow appeared to be present in the test with the fine powder, since the powder surface in the silo, when observed from above, remained
TABLE 4 Data from method 2 for determining the critical outlet width Outlet width
Comments
9.6 8.1 6.2
Flow initiated immediately the outlet
4.3 4.2
Flow initiated after knocking at hopper walls, but in s-jme cases stable arches were formed after interruptions of flow
3.2
Erratic unstable flow induced by knocking at the hopper walls. Stable arches formed after short interrup tions of flow.
by opening
smooth and horizontal until it reached transition to the hopper part.
the
CONCLUSION
As far as prediction of the critical !iopper wall slope for mass flow is concerns& the results obtained in the present investigation agree well with the results obtained by Wright [3], since in both cases an overdesign of the order of 8 - 10” was observed, and this indicates that the recommended, dotted lines in Jenike’s design graphs [l] involve this order of overdesign. For practical design purposes, however, where the possibility of uncontrolled variations in powder properties and in the wall friction must be accounted for, it might be wise, as suggested by Wright, to maintain the procedure recommended by Jenike. When it comes to the prediction of the critical outlet width for no-arching, the situation turned out to be more complicated, mainly due to difficulties in deciding how the failure loci should be extrapolated into the low normal stress region. Hence, a corresponding uncertainty as to the ;nagnitude of the unconfined failure stress, f, , which is derived on the basis of the extrapolated part of a failure locus, was the result. With the particular powder used, a fine quality Sic, this problem appeared to be particullarly exaggerated, since mosi UPthe failure loci appeared to consist of two approximately linear parts with diffzrent slopes, the slope of the low normal stress part being higher than that of the main part. Hence, two distinct alternatives for extrapolating the failure !oci were available, yielding f, values which differed by a factor of approximately two.
58
Although some experimental evidence relating the increase in the slopes of the failure loci in the low normal stress range to inadequate stress distribution in the shear cell during shear was produced, a quantitative assessment of the effect of this on the measured shear stresses at failure was not possible. If, however, the low normal stress points were totally neglected, the remaining points for most of the failure loci fell on straight lines with high degrees of precision, indicating that a linear extrapolation on the basis of these points was a sensible approach_ In the present case this approach produced an overdesign on the critical outlet width in excess of 100%. If, on the other hand, the extrapolation was based on the low normal stress point, a critical outlet width of approximately the same magnitude as the value obtained in the full-scale silo experiments was obtained. In this case, however, an adequate explanation for the distinct increase in the slope of the failure loci in the range of low normal stresses remains to be found. It should be pointed out that another, very cohesive powder being at present subjected to careful shear cell tests in our laboratory does not exhibit the very sharp transition between two parts of the failure loci as demonstrated by the Sic powder, the increase in slope occurring more gradually over a fairly wide range of normal stresses. Nevertheless, it appears that even the failure loci for these powders present considerable difficulties when it comes to the extrapolation into the region of low normal stresses. We feel that this problem of performing an adequate extrapolation of the failure loci represents an inherent difficulty in the Jenike method for design of mass flow hoppers. This problem can probably be considerably reduced by combining the shear cell tests with measurements for tensile strength, and perhaps also cohesion. In principle, a discrepancy between predicted and real critical outlet widths can arise either from an inadequate method of prcdiction, or from a lack of correspondence between the conditions for which the theory is
developed and those actually prevailing in the test silo. It is the authors’ opinion that in the present case, the full-scale silo experiments were carried out under conditions corresponding closely to the dynamic conditions presumed in the theory. Hence, it is felt that the discrepancy observed must be attributed to shortcomings in the method of prediction_ This in turn can be due either to inadequate assumptions in the theory or to shortcomings in the experimental technique used for determining the failure function, or to a combination of both factors. At present the picture is far from clear, and further systematic research aimed at resolving the problems should be encouraged. ACKNOWLEDGEMENTS
The full-scale silo was made by and the experiments on it carried out at Norton A/S, Lillesand, Norway, and the authors wish to express their gratitude to Mr. B. Aune for invaluable help. Thanks are due also to the Royal Korwegian Council for Scientific and Industrial Research for the financial support without which this investigation would not have been possible. Finally, the authors are indebted to director Dr. J.A. Andersen, CMI, for stimulating encouragement.
REFERENCES A.W. Jenike, Storageand fl-,w of solids, Bull. 123, Eng. Exp. Sta., Utah State Univ., 1964. 2 D.M. Walker, A basis for bunker design, Powder Technol., 1 (1967) 228. H. Wright, An evaluation of the Jenike bunker design method, Trans. ASME, Paper No. 72-MH-7, 1972. H. Wright, Bunker design for iron ores, Ph.D. Thesis, Univ. of Bradford, 1970. P-G. Leversen, An investigation of the flow properties of powders in silos, Thesis, Tech. Univ. of Norway, Trondheim, 1972 (in Norwegian). L. Mtirkrid, Estimation of fc and ~1 from shear cell da’s by means of regression analysis, assuming linear failure loci, CMI-Report, 1972 (in Norwegian). I