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The region of flow when discharging granular materials from bin-hopper systems
Chemical Engineering Science, 1966, Vol. 21, pp. 261-273. Pergamon Press Ltd., Oxford. Printed in Great Britain.
The region of flow when discharging granulai materials from bin-hopper systems G. C. GARDNER Central Electricity Research Laboratories,
Leatherhead,
(Received 21 April 1965; in revisedform
23 September
Surrey 1965)
Abstract-A theory is developed, equally applicable to cohesive and non-cohesive materials, which gives the shape of the boundary of the flow region in the transition from a bin section with vertical parallel walls to a hopper with converging plane walls. The theory was tested for an angular ncncohesive granular material with an angle of internal friction of 35.5”. Good agreement between experiment and theory was obtained but small consistent differences were noted. An experimental method was also devised to define the boundary of the region of fast moving material.
A BIN-HOPPERsystem comprises a bin with vertical walls joined at its base to a convergent hopper. Usually the hopper walls are straight lines in crosssectional elevation and consequently there is an abrupt transition from bin to hopper. Everyone who has studied the flow of granular material in such a system has noted that, when “mass” flow occurs (i.e. when the flow covers the whole bin cross-section), a dead, unmoving pocket of material remains in the angle between the junction of the bin and hopper walls. The shape of the boundary of the flow region is curved, becoming vertical up in the bin and tending asymptotically to a tangent to the hopper wall near the discharge port. This paper determines experimentally the shape of the boundary of the region of flow in a symmetrical plane system for a non-cohesive granular material with a known angle of internal friction. The results are compared with a theoretical boundary obtained by assuming a certain stress system within the material. The assumptions concerning the stress system will be fully explained to emphasise not only that a choice of other systems is possible but that the one chosen is reasonable and probably close to reality, as is confirmed by the experiment. The theoretical results should be equally applicable to cohesive material but no experimental test is given. It will also be shown that it is possible to dehne experimentally the boundaries of a region of fast moving material and it is probable that it is this
region which should be considered when studying the rate of discharge of granular material, as for instance in BROWN’S[l] minimum energy theorem. Material lying between the edge of the fast moving region and the boundary of the whole flow region moves very slowly but the whole flow region must be considered if it is desired to design a bin-hopper system to avoid any dead material. The shape of the boundary of the flow region has previously been investigated by KVAPIL [8] and O’CALLAGHAN[9]. Kvapil empirically fitted a section of an ellipse but its dimensions were not correlated with measurable physical properties of the material, such as the angle of internal friction and cohesion. O’Callaghan followed the ideas used in soil mechanics for the failure surfaces of embankments and fitted the logarithmic spiral r = r&e ‘an1
(1)
where r and 6 are polar coordinates, r. is a constant and q is the angle of internal friction of the material. For shallow beds of grain the spiral fitted O’Callaghan’s results quite well, it if were chosen to pass through the edge of the discharge slot and was vertical at the top free surface of the grain. The origin of the spiral and the constant r. had to be determined by experiment. For deep beds of grain the spiral was chosen to fit the experimental results as well as to pass through the edge of the discharge slot and to become tangent to the vertical bin wall. In this case the logarithmic spiral did not agree very well with experiment. The theory and experiment
261
G. C. GARDNER The flow region is divided internally into three parts. Firstly, above OB the stress system is that for plastic equilibrium in a bin well below a top free surface. Secondly OCDO encloses a region in which the so-called radial stress system for plastic equilibrium obtains. This system is calculated relatively easily by numerical means and has been shown by JENIKE [5] to be approached near the apex of a wedge. Lastly the region enclosed by OBCO has to be computed using stresses on OB and OC as boundary conditions and gives the required shape of the curved part of the flow region’s boundary BC. The material throughout the flow region is assumed to be in a state of plastic equilibrium, obeying Coulomb’s failure criterion; i.e. at any point within the material failure is occurring upon two planes at an angle of (7-c/2- r,) to each other. The stresses causing failure are given by
of the present paper apply to deep beds of granular material and the theory defines the boundary of the region of flow in terms of the angle of internal friction and no other empirical constant. THE ASSUMEDSTRESSSYSTEM
Figure 1 shows the boundary ABCD of the flow region. Stresses in the plane of observation are assumed to be uninfluenced by stresses parallel or normal to the plane. GARDNER [4] has given theoretical reasons indicating that, if flow tends to diverge very slightly rather than to converge in planes normal to the one of interest and if any walls parallel to the plane considered are smooth, the assumption is justified. Experiment also supports this conclusion. AB of the boundary is parallel to the vertical bin wall and CD is parallel to a plane hopper wall of chosen angle to the vertical. BC is the curved part of the boundary and is the subject of the investigation.
]r] = c + fr tan y
(2)
where z and cr are shear and normal compressive stress upon a plane, c is the constant cohesion and 17 is the constant angle of internal friction. Thus there are two families of failure or slip-lines covering the flow region. In particular OB and OC in Fig. 1 are slip lines, since it is mathematically convenient to choose them to subdivide the flow region. Stresses along and above OB
FIG. 1.
The subdivisionof the stress field.
It should be noted that there is no physical necessity for plastic equilibrium to exist in the bin. The flow does not converge and a rigid mass of material, failing only at the wall AB, is possible. The material passes, however, into a converging region where plastic equilibrium must obtain and thus, for simplicity, it is assumed that material is in a plastic state in the bin as well. The alternative is to assume some rigid stress system in the bin, which itself would be arbitrarily chosen, and then calculate stresses in a transition region, terminating at a line along which the material became plastic. It is well known-see, for example, DELAPLAINE [3]-that, since granular materials can support a shear stress, stresses become substantially invariant with depth at between three and five bin span widths below a top free surface. It is assumed 262
The region of flow when discharging granular materials from bin-hopper
0, + c cot q = p[l + sin 1 cos g] = yL cot 11
that this condition applies here and the theory is limited to that case. Thus the equations of static stress equilibrium
ac, a7 ax+%= ac _2+t ay
(3)
PO =
yAcot Pj 1 + sin rj
can be integrated to give
I
/..\2
-
(5)
z= -yx
(11)
Substituting p from Eq. (10) into Eqs. (6) and (9) gives an equation in g with the solution
(4)
0, = constant
(10)
If p = p. at 0 in Fig. 1 and g is simultaneously zero, for reasons of symmetry, Eq. (10) yields
o
c;x =-’
systems
cos g =
(6)
where x is the horizontal coordinate measured from the axis of symmetry, y is the vertical coordinate measured upwards, y is the material density and z = 0 when x = 0 for reasons of symmetry. Now let the stress parameter p and the angle g be defined by the Coulomb-Mohr diagram of Fig. 2. g is twice the angle between the x direction and the principal stress direction. Then
X 0
,..\2
sin q + cos2 q
Jl-(J x2 cos2 rj + 0-; sin2 y \A/ (12)
Substituting Eq. (12) in Eq. (10) and using Eq. (11) gives
6, = p[l + sin q cos g] - c cot rj
(7)
(13)
fry = p[l - sin q cos g] - c cot ‘1
(8) (9)
Equations (12) and (13) are sufficient to define the stresses with respect to x. To find the shape of the slip-line OB we use the definition of g and referring to Fig. 2
z = p sin rj sin g
Also let x = A at the wall AB. Then from Eqs. (5) and (7) and because the point (crx,,z) at x = L is on the yield line of Fig. 2, since failure occurs parallel with the wall within the material at the wall,
Yield
& --tan dx
lin
FIG. 2.
Coulomb-Mohr
263
diagram.
[ 1 !+p,
7cr
p=4-2
(14)
G. C. GARDNER
be considered as an arbitrary stress applied to a material with weight at some distance from the apex. Evidently the arbitrary stress will disappear rapidly and only the radial stress system for the material with weight will be left.
The radial stress system in region OCDO The radial stress system and some solutions for it have been described by SOKOLOVSKI[lo] and JENIKE [5], although they only noted its application to the flow of cohesionless material. It describes plastic stress equilibrium in a wedge. If polar coordinates are used, with the apex of the wedge as the origin, the system has the properties that along one radial ray the stress parameter p is proportional to r and the slip-lines have a constant inclination to the ray or, in other words, g is constant. One side of the wedge is the envelope to one family of slip-lines and the other side is the envelope to the other family. One such slip-line is OC in Fig. 1, though it should be noted that it should extend to the apex D. The numerical method of computation, however, makes it impossible to distinguish the slip-line from a straight line between C and D. Radial stress solutions for gravity flow can be found for wedges with sides at an angle of more than q to the horizontal. The theory therefore infers that, for hopper walls at less than q to the horizontal, the tangent to the region of flow at the discharge slot will itself be q to the horizontal. Referring to Fig. 1, it will be noticed that it has been assumed that the stress parameter p at the axis of symmetry is constant above 0 and decreases linearly to zero from 0 to D. Such an abrupt change in the gradient of p at 0 is unlikely but can be justified as an approximation if it can be shown that the radial stress system is rapidly attained in a wedge. JOHANSON[7] has calculated stresses for a wedge of material with a top free surface and has shown that the radial stress system is indeed achieved close to the top free surface. Another way of looking at the problem is to start by considering the radial stress system for a material without weight. SOKOLOVSKI [lo] has shown that in this case the stress parameter along a radial ray varies as r” where n is a constant which gets larger as the included angle of the wedge gets smaller. Now q typically equals 40” for a granular material and then the smallest value of n is 4.3, applying to a hopper out of whose apex material will flow under gravity. Now the stress system for a material without weight can, as an approximation,
Calculation of the stress system in region OBCO The shape of and stresses along slip-lines OB and OC are known. SOKOLOVSKI[lo], for example, describes how the stresses in the region OBCO can then be calculated using the method of characteristics. Referring to Fig. 3, OB is one of the 5 E-lines
FIG. 3.
Calculation of the stress field.
family of slip-lines and OC is one of the E family of slip-lines.
dy
on 6 slip-lines - = tan dx
(15)
dY on E slip-lines - = tan ax
(16)
r and E functions are defined by
264
07)
,,!$!?lnP
( 1_!2 PO
(18)
The region of flow when discharging granular materials from bin-hopper systems
Further
d5 = -
on t slip-lines -
dx
on E
de
slip-lines -
dx
=
In Fig. 3, if all properties at points a and b are known, it is possible by writing the approximation
this manner the stresses in the whole field can be determined by generating < and E lines from the boundaries OB and OC. As a result of the numerical computations using a digital computer it is found, due to the inaccuracy of the method near the boundary of the flow region BC, that the E lines double back into the flow region whereas they should continue down to the hopper apex at D. The doubling back is sharp, however, and it is easy to define an approximation to the boundary BC which is close to the real envelope of the E lines. REXJLTSOFTHE THEORETICAL CALCULATKJNS
& At dx=?G and so on to determine the properties at point c. In
The results of the calculations are given in Tables l-4 for materials with angles of internal friction 30”, 35*5”, 40” and 50”. The tables define
Table 1. Theoretical edge of moving region for q = 30” Hopper included half-angle 10.33”
5.05”
18.13”
30.1”
60.0”
36.8”
(x/Q
(Yl&
(x/h)
(Y/X)
(x/h)
(Yl&
@I~)
(Y/h)
@I~)
0.903 0.918 0.934 0.956 0.965 0.983 0.993 0.999
-2.195 - 2.057 - 1903 - 1641 - 1.514 - 1.173 -0.919 -0.710
0.791 0.834 0.873 0.915 0.935 0.961 0.981 0.987 0.992 0.997
-2.198 -2WO - I.791 - 1.519 - 1.350 - 1.076 -0.853 -0.716 -0.656 -0.565
0.675 0.776 0.835 0.891 0.937 0.962 0.974 0.983 0.991 l@OO
-2.142 - 1.843 - 1.621 - 1.359 - 1.070 -0.880 -0.761 - 0648 -0.538 -0.380
0.581 0.714 0.812 0.906 0.942 0.953 0.968 0.983 0.995
- 2.036 - 1.753 - 1.474 - 1.093 -0.895 -0.817 -0.702 -0.551 -0403
0.389 0.661 0.790 0,886 0.941 0.956 0.965 0.980 0.994
(Y/A> -2.214 -1.794 -1.458 -1.129 -0.892 -0.758 -0.691 -0.558 -0.392
WV
(Yl&
0.273 0.558 0.734 0.861 0.920 0.936 0.958 0.972 0.982 0.990
-2.187 -1.859 -1,524 - 1.151 -0.914 -0.832 - 0.695 -0.584 -0.501 -0.416
Table 2. Theoretical edge of moving region for rl = 35.5” Hopper included half-angle 5.81”
12.52
19.07”
@I~)
(Yl&
(xl&
(Yl&
0.879 0.895 0.917 0.943 0.964 0.983 0.994 0.999
- 2.487 -2.362 -2.176 - 1.894 - 1,630 -1.226 -0.998 -0.809
0.793 0.821 0.841 0.883 0.910 0.930 0.967 0.982 0.992 0.999
-2.321 -2,193 - 2.096 - 1.852 -1.667 -1.510 - 1.141 -0.943 -0.769 -0.612
(Y/x,
(xl&
(Y/x>
(xl&
-2.511 -2.306 -2.023 - 1,794 -1.515 -1.204 - 1.003 -0.879 - 0.763 - 0.599
0.467 0.653 0.754 0.840 0.918 0.946 0.952 0.967 0.919 0.996
-2.558 -2.193 - 1.923 - 1.618 - 1.236 - 1.059 -0.972 -0.888 -0.764 - 0.529
0.208 0.512 0.683 0.806 0898 0.936 0.951 0.968 0.982 0.994
W) 0.634 0.709 0.793 0.847 O+OO 0.943 0.966 0.916 0.986 0.996
54.5”
31.8’
265
G. C. GARDNER
Table 3. Theoretical edge of moving region fbr q = 40” Hopper included half-angle 660”
14.82
(xl&
(Y/4
0.892 0909 0.935 0.948 0972 0.985 0.994 0.999
- 2.494 -2.349 -2.106 - 1.934 - 1.551 - 1.281 - 1.072 -0.898
(-4) 0.684 0.749 0.798 0.856 0.886 0.925 0.960 0.976 0.987 0.998
25.1”
(Y/x) -2.813 -2.585 -2384 - 2.093 -1.918 - 1.632 - 1.294 -1.107 -0.941 -0.724
(xl&
(Y/a
0.517 0.653 0.750 0.816 0.880 0.932 0.961 0.912 0.984 0.996
50.0
35.6
-2.888 - 2.567 - 2.273 -2.021 -1.719 - 1.381 - 1.140 -1.028 -0.870 -0.631
m
(Y/a
0.292 0532 0666 0.771 0.852 0918 0.963 0.976 0.986 0.994
-3.100 -2.715 -2m4 - 2.086 -1.769 -1404 - 1.107 -0.932 -0.803 -0641
w 0.154 0442 0.614 0.744 0.828 0.910 0.944 0.954 0.964 0.972 0.982 0.995
(Y/h) -3.165 -2.808 -2.471 -2.125 -1.822 -1.418 -1.200 -1.115 -1.031 -0.949 -0.830 - 0642
Table 4. Theoretical edge of moving region for q = 50” half-angle
Hopper included 9.05”
40
16.3”
(xl&
(Y/a
0.733 0.860 0.890 0+08 0.922 0.952 0.982 0.990 0.996
-3.648 -2.979 -2.743 -2.577 -2QI40 - 2.058 - 1.628 - 1.355 -1.193
(xl& 0.573 0718 0.785 0.828 0.888 0.953 0.972 0.979 0.989 0.996 0.999
the wall boundary BC in terms of X = (x/A) and Y = b/A) where 1 is the half-width of the bin section. Different wall boundaries are given for different included hopper half-angles. It is noticed that the cohesion does not occur as a parameter in the results but the results are equally applicable to materials with cohesion as without it. The results for an angle of internal friction of 35.5” are drawn in Fig. 4 to illustrate that the shape of the curved boundary BC becomes progressively less curved as the included hopper angle becomes smaller.
(Y/3
c4v
-3.791 - 3.270 -2.974 -2.731 -2.349 - 1.746 - 1.506 - 1.396 - 1.200 - 1.029 -0.952
0.195 0.267 0460 0,665 0.727 0.801 0.877 0,931 0,962 0.973 0.982 0.997
-4.217 -4.112 - 3.732 -3.169 -2.944 - 2.626 -2.211 - 1.800 - 1.495 - 1.352 - 1.218 -0.923
THE EXPERIMENT
The theory has been concerned with a plane problem which, it has been explained, can be studied experimentally if the material tends to diverge very slightly rather than to converge in planes normal to the plane of interest and if the containing walls parallel to the plane of observation are smooth. Both these requirements are met in the apparatus. The theory has also been concerned with a stress system in which one family of slip-lines is tangent to the region of flow or the material fails within
266
The region of flow when discharging granular materials from bin-hopper X/X
Hopper
FIG. 4.
Theoretical
included
shape of edge of moving region. 7) = 35.5”.
itself at the boundary. This situation is attainable experimentally if the walls normal to the plane of observation are rough so that the shear stress, which would give slip of the material against the wall, is greater than for failure within the material itself. To make the walls rough a layer of the granular material was glued to them. This certainly makes the wall-material failure condition stronger than the material-material condition since the ability of the particles to rotate has been removed. There is, however, quite likely a transition layer adjacent to the walls in which the material strength properties change from one condition to the other and which may cause a difference between theory and practice. Such changes in material properties may indeed occur wherever there is a gradient of velocity normal to the direction of flow. The experimental
apparatus
The apparatus is illustrated in Fig. 5. The binhopper profile was provided by two vertical wooden walls, bevelled at the bottom so that two shorter plane walls could be fitted at any angle from 0” to
systems
65” to the horizontal immediately beneath them. The two sloping walls formed the hopper and their position could be adjusted to give a sharp edged slot discharge of any chosen width. The other two walls of the container were of Perspex, which is sufficiently smooth and allowed observation of the material. These walls were not quite vertical since the two vertical wooden walls, to which they were screwed, changed in width over their 4 ft height from 3 in. at the top to 3* in. at the bottom, thus avoiding any tendency to convergent flow normal to the plane of observation. The span between the wooden walls of the bin was 6 in. The internal wooden surfaces of the apparatus were coated with impact adhesive followed by a layer of the experimental granular material. The rough surface so formed proved durable for the number of experiments carried out. Discharge through the slot at the bottom of the hopper could be prevented by means of a hinged trap set about $ in. below the slot and held in place by an easily removed pin. When the pin was pulled out the trap swung free of the slot. The trap could also be rapidly swung back into place and refastened without disturbing the material in the bin-hopper system. The experimental material was an angular calcined flint named “Durite” of from & in. to & in. in size. It did not change shape or dust after passing it many times through the apparatus. It was also porous and thus easily dyed a deep blue by ink. It was determined, by means of a shear cell, that the “Durite” was cohesionless and had an angle of internal friction of 35.5”. Experimental
method
The system was filled with horizontal layers of “Durite” as shown in Fig. 5. The layer of undyed “Durite” was poured in and gently levelled with a stick. Then dyed material was poured against the Perspex wall and not over the whole cross section to limit the quantity used. The dyed layers were spaced more closely at the bottom of the apparatus than at the top. Material was discharged from the slot until the top surface in the bin was about 23 ft above the slot. 267
G. C. GARDNER
Wooden
side walls
/ Perspex front and back walls
Dyed
layers
et f
Hopper section in horizontal position
/ Swinqinq trap stop and start Dircharqe’slot
FIG. 5.
Experimental
268
apparatus.
The region of flow when discharging granular materials from bin-hopper
The slot width was about t in. which gave a convenient discharge rate of 5 cm3/sec and variation of the rate did not alter the experimental result. The bin was topped up again, without layering, and more material was discharged. The procedure was continued until the flow pattern was fully developed and the remaining outer edges of the horizontal dyed lines suffered no further erosion. Three methods were then used to determine the boundary of the moving material: 1. The position of the edge of the dyed layers nearest the axis of symmetry was measured in terms of x and y. Thus from one run four sets of measurements were obtained, two from each Perspex face. Each set was plotted individually together with the position of the edge of the slot and a smooth curve was drawn and extrapolated the small distance to the axis of symmetry. The points were then replotted using the intersection with the axis of symmetry as the origin and reducing the co-ordinates to (x/A) and (V/A). Only then were the results compared with the theoretical curve. The data obtained in this manner are given in Figs. 6 and 7. 2. Photographic time exposures of 3 set to 1 min duration were taken of the flowing material. The defined flow region was identical to that given by method 1 as is illustrated by Fig. 13. 3. A 6 in. depth of dyed material was poured into the top of the bin and run right through the apparatus while keeping the bin topped up with undyed material. It left a trail of blue material, which is clearly shown in Fig. 12 and which ran the whole height of the binhopper system. The trail lay inside the edge of the flow region defined by the horizontal blue layers, and, although it would all have been removed if more material had been allowed to flow through the apparatus, it nonetheless gave a clear definition of the fastest moving flow region. Measurements were taken from the outside edge of the trail and treated in the same manner as for method 1. The final results, in reduced co-ordinates, are plotted in Figs. 8 and 9. 269
l
systems
o m A
Position of the ends sets of dyed layers
X
Edqe of
Theoretical curve
discharqe
of
four
slot
‘I = 4o” ‘1 = 35*s”,
FIG. 6.
Shape of edge of whole flow region determined from horizontal dyed layers for hopper with horizontal Walls.
G. C. GARDNER
DOrn0
Position sets of
X
Edqe
of
of the ends of dyed layers
discharqe
four
I
slot
l
I% -
I
o D 6
x
Measurements trails
left
Edqe
of
from
four
by iluq discharqe
slot
C C
2-
I Q
n l
I-
h
7 .
m O! / =
.i
0 8
4 /
b
< i C1
Shape of edge of whole flow region determined from horizontal dyed layers for hopper with walls at 58” to the horizontal.
‘
x
FIG. 7.
FIG. 8. Shape of edge of fast moving region given by slug of blue material in hopper with horizontal walls.
270
The region of flow when discharging granular materials from bin-hopper
l 0
m 4
X
Measurements from trails left by $1~19 Edqe
of
discharqe
Experiments were carried out for hopper wall angles of 0” and 58” to the horizontal. Theoretically, the results should not vary for hopper angles from 0” to 355” to the horizontal, where 35.5” is the angle of internal friction.
four
Experimental observations slot
I 29
Shape of edge of fast moving region given by 9. slug of blue material in hopper with walls at 58” to horizontal.
FIG.
systems
Before the results are compared with theory, some experimental observations are important. Figure 10 shows the system shortly after flow had started. The flow region is a long ellipsoidal shape located on the axis of symmetry. As the flow region develops, this ellipsoidal region broadens and penetrates further up the bin section until the top free surface is reached. The flow region then continues to broaden but the sides, well above the discharge slot, become vertical and parallel. Measurementsaccordingtomethodl thengiveresults identical to those shown in Figs. 6 and 7 for the fully developed flow. The curve in the real x, y coordinates instead of the reduced co-ordinates is, of course, smaller by the difference in the scale factor 1. If only the bottom half of the bin is layered and undyed material is poured into the top half, the flow develops in the same manner except that, when the flow region reaches the unlayered material, it spreads rapidly. It appears that, even though the material was only very gently spread level by the stick in layering, a structure has been built into the layered material which makes it stronger than material which either has recently flowed or is in a state of flow. The flow in the layered material spreads very slowly to cover the whole possible flow region. When the fully or partly developed flow is stopped, as shown in Figs. 10-12, and then started again, the flow region continues to develop from the point at which it stopped, indicating that the layering operation is the cause of the observed development of flow. The slow spread of the flow region illustrates what may happen in bunkers that have been closed down for a period. It also introduces an element of doubt into these experiments that have previously been performed with layered materials and have been concerned only with initial deformations. An example of this is JOHANSON’S [7] experiments to co&m the JENIKEand SHIELD[6] flow theory. KVAPIL [8] also developed a theory of bin-hopper 271
G. C. GARDNER
design in which he observed the elongated ellipsoidal shape and used its dimensions as design data. Another observation of importance is that just above the discharge slot there is a tongue of material moving much faster than the surrounding material. This has been observed by others, for instance BROWN and HAWKSLEY [2]. The tongue is seen in some of the time exposures but most clearly in Fig. 11 where the flow was stopped when the blue plug of material was just beginning to discharge. The blue tongue moved rapidly ahead of the rest of the blue material. Lastly it is noted that the edge of the flow region never reached the bin walls completely. The walls were sufficiently rough to ensure that the material failed within itself and not with respect to the wall. Discussion of the experimental results Figures 6 and 7 show the experimental results obtained by method 1 where the position of the ends of the dyed layers was measured. This defines the complete flow region. The theoretical curves for angles of internal friction 30”, 35.5” and 40” are also plotted on Fig. 6 for the hopper with a wall angle to the horizontal of 0”. The theoretical line for the measured angle of internal friction 355” fits the data best and indeed fits it very well. It is, however, noticeable that the experimental points tend to cross the theoretical line. At the base of the hopper they correspond better with the theoretical line corresponding to an angle of internal friction of 40” and near the bin section with the theoretical line corresponding to an angle of internal friction of 30”. It has been indicated that differences between theory and experiment may be due to the choice of the stress system, the variation in the strength properties of the material, especially near the boundary of the flow region, and lastly due to neglect of kinetic stresses, especially near the discharge slot where the fast moving tongue of material occurs. Figure 7 is for the hopper wall at an angle of 58” to the horizontal. The same comments apply but the correlation is equally good. Figures 8 and 9 show the results obtained by method 3 from the trail of the plug of dyed material. Where the angle of the hopper wall to the horizontal is 0”, the correlation appears to be quite good near
the bin section but diverges considerably from theory near the discharge slot. The discrepancy is even greater when the angle of the hopper wall to the horizontal is 58”. In both cases the edge of the fast-moving core ilow, defined by this experimental method, appears to be a straight line through the origin for the greater part of the hopper flow region. It must be concluded that the theory does not give an adequate description of the measurements by this method. It should be noted that the extent of the fastmoving core flow is probably of importance in problems concerning the rate of discharge of materials and should be used in say the theory of BROWN [l]. On the other hand the shape of the whole flow region will be important in the design of bin-hopper systems to ensure that there are no “dead” spaces which could prove troublesome with respect to cohesive materials. It is re-emphasised that the theory is applicable to cohesive material but has only been tested experimentally for a non-cohesive material. Experiments with cohesive materials are desirable and might determine whether the angle of internal friction or JENIKE’S[5] apparent angle of internal friction are important. CONCLUSIONS
A theory has been developed, equally applicable to cohesive and non-cohesive materials, which gives the shape of the flowing region in the transition from a bin section with vertical parallel walls to a hopper of converging plane walls. The results are given in Tables l-4 and in terms of reduced co-ordinates (x/A) and b/A) where 1 is the half span of the bin section. The theory was tested for an angular non-cohesive material with an angle of internal friction of 35.5”. The comparison of theory with experiment is good but it is noted that there are consistent differences which, although they may be of negligible importance in defining the transitional wall shape, may be important in other problems concerning the flow of granular materials such as the rate of discharge through the discharge slot. Near the outlet slot the difference may be attributed to the influence of a tongue of fast-moving material projecting a small 272
The region of flow when discharging granular materials from bin-hopper
way up into the mass. This tongue is well known under free discharge conditions. Near the bin section the difference may be due either to the change of properties of the material between the fast-moving
core
and
the flow boundary
or to the
of particular stress fields in the theory. These stress fields are fully discussed. The experiment also showed that the shape of the fast moving core could be well defined by allowing a plug of dyed material to pass through the apparatus.? It left a trail at its edge which lay well within the boundaries of the complete flow region. Results obtained by this method are presented.
systems
He also devised the method of deiining the extent of the fast moving region using the blue slug of material. Mr. J. C. RICHARDSof B.C.U.R.A. provided the calcined
flint and measuredits ande of internal friction.
The work was carried o;t at the Central Electricity Research Laboratories and the paper is published by permission of the Central Electricity Generating Board.
assumption
AcknowledgementsMr. D. LATHAMcarried out the experimental part of the work during his undergraduate vacation.
7 It has recently come to the writer’s notice that J. B. MARTIN and J. C. RICHARDS(J. Science Technology 1965 11 31) found this method convenient with respect to conical hoppers.
NOTATION Cohesion Twice the angle between the x direction and the g direction of action of the major principal stress n Constant exponent P Stress parameter defined by Fig. 2 PO Constant value of stress parameter r Radial co-ordinate X Horizontal co-ordinate Vertical co-ordinate Material density r’ Stress characteristic parameter defined by Eq. (18) & Angle of internal friction Angular co-ordinate 3 Half-width of flow region in the bin h (n/4 - 77/2) P u Normal compressive stress Shear stress Stress characteristic parameter defined by ,Eq. (17) i C
REFERENCES
111 BROWNR. L., Nature, Lond. 1961 191458. PI BROWNR. L. and HAWKSLEY P. G. W., Fuel, Lond. 1947 26 159. DELAPLAINEJ. D. A.1.ChX.X
1956 2 127.
;:; GARDNERG. C., Chem. Engng Sci. 1964 19 283. PI JENIKEA. W., Bulletin No. 108 of Utah Engineering Experiment Station 1961. 161 JEN~E A. W. and SHIELDR. T., J. appl. Mech. 1959 81B 599. t71 181 t91 1101
C
JOHANSONJ. R., J. appl. Mech. 1964 31 499. KVAPIL R., Theorie der Schuttgutbewegung V.E.B. Verlag Technik, Berlin, 1959. O’CALLAGHANJ. R., J. agric. Engng Res. 1960 5 200. SOKOLOVSKI V. V., Statics of Soil Media pp. 14 and 139, Butterworths, 1960.
273
The region of flow when discharging granulai materials from bin-hopper systems G. C. GARDNER Central Electricity Research Laboratories,
Leatherhead,
(Received 21 April 1965; in revisedform
23 September
Surrey 1965)
Abstract-A theory is developed, equally applicable to cohesive and non-cohesive materials, which gives the shape of the boundary of the flow region in the transition from a bin section with vertical parallel walls to a hopper with converging plane walls. The theory was tested for an angular ncncohesive granular material with an angle of internal friction of 35.5”. Good agreement between experiment and theory was obtained but small consistent differences were noted. An experimental method was also devised to define the boundary of the region of fast moving material.
A BIN-HOPPERsystem comprises a bin with vertical walls joined at its base to a convergent hopper. Usually the hopper walls are straight lines in crosssectional elevation and consequently there is an abrupt transition from bin to hopper. Everyone who has studied the flow of granular material in such a system has noted that, when “mass” flow occurs (i.e. when the flow covers the whole bin cross-section), a dead, unmoving pocket of material remains in the angle between the junction of the bin and hopper walls. The shape of the boundary of the flow region is curved, becoming vertical up in the bin and tending asymptotically to a tangent to the hopper wall near the discharge port. This paper determines experimentally the shape of the boundary of the region of flow in a symmetrical plane system for a non-cohesive granular material with a known angle of internal friction. The results are compared with a theoretical boundary obtained by assuming a certain stress system within the material. The assumptions concerning the stress system will be fully explained to emphasise not only that a choice of other systems is possible but that the one chosen is reasonable and probably close to reality, as is confirmed by the experiment. The theoretical results should be equally applicable to cohesive material but no experimental test is given. It will also be shown that it is possible to dehne experimentally the boundaries of a region of fast moving material and it is probable that it is this
region which should be considered when studying the rate of discharge of granular material, as for instance in BROWN’S[l] minimum energy theorem. Material lying between the edge of the fast moving region and the boundary of the whole flow region moves very slowly but the whole flow region must be considered if it is desired to design a bin-hopper system to avoid any dead material. The shape of the boundary of the flow region has previously been investigated by KVAPIL [8] and O’CALLAGHAN[9]. Kvapil empirically fitted a section of an ellipse but its dimensions were not correlated with measurable physical properties of the material, such as the angle of internal friction and cohesion. O’Callaghan followed the ideas used in soil mechanics for the failure surfaces of embankments and fitted the logarithmic spiral r = r&e ‘an1
(1)
where r and 6 are polar coordinates, r. is a constant and q is the angle of internal friction of the material. For shallow beds of grain the spiral fitted O’Callaghan’s results quite well, it if were chosen to pass through the edge of the discharge slot and was vertical at the top free surface of the grain. The origin of the spiral and the constant r. had to be determined by experiment. For deep beds of grain the spiral was chosen to fit the experimental results as well as to pass through the edge of the discharge slot and to become tangent to the vertical bin wall. In this case the logarithmic spiral did not agree very well with experiment. The theory and experiment
261
G. C. GARDNER The flow region is divided internally into three parts. Firstly, above OB the stress system is that for plastic equilibrium in a bin well below a top free surface. Secondly OCDO encloses a region in which the so-called radial stress system for plastic equilibrium obtains. This system is calculated relatively easily by numerical means and has been shown by JENIKE [5] to be approached near the apex of a wedge. Lastly the region enclosed by OBCO has to be computed using stresses on OB and OC as boundary conditions and gives the required shape of the curved part of the flow region’s boundary BC. The material throughout the flow region is assumed to be in a state of plastic equilibrium, obeying Coulomb’s failure criterion; i.e. at any point within the material failure is occurring upon two planes at an angle of (7-c/2- r,) to each other. The stresses causing failure are given by
of the present paper apply to deep beds of granular material and the theory defines the boundary of the region of flow in terms of the angle of internal friction and no other empirical constant. THE ASSUMEDSTRESSSYSTEM
Figure 1 shows the boundary ABCD of the flow region. Stresses in the plane of observation are assumed to be uninfluenced by stresses parallel or normal to the plane. GARDNER [4] has given theoretical reasons indicating that, if flow tends to diverge very slightly rather than to converge in planes normal to the one of interest and if any walls parallel to the plane considered are smooth, the assumption is justified. Experiment also supports this conclusion. AB of the boundary is parallel to the vertical bin wall and CD is parallel to a plane hopper wall of chosen angle to the vertical. BC is the curved part of the boundary and is the subject of the investigation.
]r] = c + fr tan y
(2)
where z and cr are shear and normal compressive stress upon a plane, c is the constant cohesion and 17 is the constant angle of internal friction. Thus there are two families of failure or slip-lines covering the flow region. In particular OB and OC in Fig. 1 are slip lines, since it is mathematically convenient to choose them to subdivide the flow region. Stresses along and above OB
FIG. 1.
The subdivisionof the stress field.
It should be noted that there is no physical necessity for plastic equilibrium to exist in the bin. The flow does not converge and a rigid mass of material, failing only at the wall AB, is possible. The material passes, however, into a converging region where plastic equilibrium must obtain and thus, for simplicity, it is assumed that material is in a plastic state in the bin as well. The alternative is to assume some rigid stress system in the bin, which itself would be arbitrarily chosen, and then calculate stresses in a transition region, terminating at a line along which the material became plastic. It is well known-see, for example, DELAPLAINE [3]-that, since granular materials can support a shear stress, stresses become substantially invariant with depth at between three and five bin span widths below a top free surface. It is assumed 262
The region of flow when discharging granular materials from bin-hopper
0, + c cot q = p[l + sin 1 cos g] = yL cot 11
that this condition applies here and the theory is limited to that case. Thus the equations of static stress equilibrium
ac, a7 ax+%= ac _2+t ay
(3)
PO =
yAcot Pj 1 + sin rj
can be integrated to give
I
/..\2
-
(5)
z= -yx
(11)
Substituting p from Eq. (10) into Eqs. (6) and (9) gives an equation in g with the solution
(4)
0, = constant
(10)
If p = p. at 0 in Fig. 1 and g is simultaneously zero, for reasons of symmetry, Eq. (10) yields
o
c;x =-’
systems
cos g =
(6)
where x is the horizontal coordinate measured from the axis of symmetry, y is the vertical coordinate measured upwards, y is the material density and z = 0 when x = 0 for reasons of symmetry. Now let the stress parameter p and the angle g be defined by the Coulomb-Mohr diagram of Fig. 2. g is twice the angle between the x direction and the principal stress direction. Then
X 0
,..\2
sin q + cos2 q
Jl-(J x2 cos2 rj + 0-; sin2 y \A/ (12)
Substituting Eq. (12) in Eq. (10) and using Eq. (11) gives
6, = p[l + sin q cos g] - c cot rj
(7)
(13)
fry = p[l - sin q cos g] - c cot ‘1
(8) (9)
Equations (12) and (13) are sufficient to define the stresses with respect to x. To find the shape of the slip-line OB we use the definition of g and referring to Fig. 2
z = p sin rj sin g
Also let x = A at the wall AB. Then from Eqs. (5) and (7) and because the point (crx,,z) at x = L is on the yield line of Fig. 2, since failure occurs parallel with the wall within the material at the wall,
Yield
& --tan dx
lin
FIG. 2.
Coulomb-Mohr
263
diagram.
[ 1 !+p,
7cr
p=4-2
(14)
G. C. GARDNER
be considered as an arbitrary stress applied to a material with weight at some distance from the apex. Evidently the arbitrary stress will disappear rapidly and only the radial stress system for the material with weight will be left.
The radial stress system in region OCDO The radial stress system and some solutions for it have been described by SOKOLOVSKI[lo] and JENIKE [5], although they only noted its application to the flow of cohesionless material. It describes plastic stress equilibrium in a wedge. If polar coordinates are used, with the apex of the wedge as the origin, the system has the properties that along one radial ray the stress parameter p is proportional to r and the slip-lines have a constant inclination to the ray or, in other words, g is constant. One side of the wedge is the envelope to one family of slip-lines and the other side is the envelope to the other family. One such slip-line is OC in Fig. 1, though it should be noted that it should extend to the apex D. The numerical method of computation, however, makes it impossible to distinguish the slip-line from a straight line between C and D. Radial stress solutions for gravity flow can be found for wedges with sides at an angle of more than q to the horizontal. The theory therefore infers that, for hopper walls at less than q to the horizontal, the tangent to the region of flow at the discharge slot will itself be q to the horizontal. Referring to Fig. 1, it will be noticed that it has been assumed that the stress parameter p at the axis of symmetry is constant above 0 and decreases linearly to zero from 0 to D. Such an abrupt change in the gradient of p at 0 is unlikely but can be justified as an approximation if it can be shown that the radial stress system is rapidly attained in a wedge. JOHANSON[7] has calculated stresses for a wedge of material with a top free surface and has shown that the radial stress system is indeed achieved close to the top free surface. Another way of looking at the problem is to start by considering the radial stress system for a material without weight. SOKOLOVSKI [lo] has shown that in this case the stress parameter along a radial ray varies as r” where n is a constant which gets larger as the included angle of the wedge gets smaller. Now q typically equals 40” for a granular material and then the smallest value of n is 4.3, applying to a hopper out of whose apex material will flow under gravity. Now the stress system for a material without weight can, as an approximation,
Calculation of the stress system in region OBCO The shape of and stresses along slip-lines OB and OC are known. SOKOLOVSKI[lo], for example, describes how the stresses in the region OBCO can then be calculated using the method of characteristics. Referring to Fig. 3, OB is one of the 5 E-lines
FIG. 3.
Calculation of the stress field.
family of slip-lines and OC is one of the E family of slip-lines.
dy
on 6 slip-lines - = tan dx
(15)
dY on E slip-lines - = tan ax
(16)
r and E functions are defined by
264
07)
,,!$!?lnP
( 1_!2 PO
(18)
The region of flow when discharging granular materials from bin-hopper systems
Further
d5 = -
on t slip-lines -
dx
on E
de
slip-lines -
dx
=
In Fig. 3, if all properties at points a and b are known, it is possible by writing the approximation
this manner the stresses in the whole field can be determined by generating < and E lines from the boundaries OB and OC. As a result of the numerical computations using a digital computer it is found, due to the inaccuracy of the method near the boundary of the flow region BC, that the E lines double back into the flow region whereas they should continue down to the hopper apex at D. The doubling back is sharp, however, and it is easy to define an approximation to the boundary BC which is close to the real envelope of the E lines. REXJLTSOFTHE THEORETICAL CALCULATKJNS
& At dx=?G and so on to determine the properties at point c. In
The results of the calculations are given in Tables l-4 for materials with angles of internal friction 30”, 35*5”, 40” and 50”. The tables define
Table 1. Theoretical edge of moving region for q = 30” Hopper included half-angle 10.33”
5.05”
18.13”
30.1”
60.0”
36.8”
(x/Q
(Yl&
(x/h)
(Y/X)
(x/h)
(Yl&
@I~)
(Y/h)
@I~)
0.903 0.918 0.934 0.956 0.965 0.983 0.993 0.999
-2.195 - 2.057 - 1903 - 1641 - 1.514 - 1.173 -0.919 -0.710
0.791 0.834 0.873 0.915 0.935 0.961 0.981 0.987 0.992 0.997
-2.198 -2WO - I.791 - 1.519 - 1.350 - 1.076 -0.853 -0.716 -0.656 -0.565
0.675 0.776 0.835 0.891 0.937 0.962 0.974 0.983 0.991 l@OO
-2.142 - 1.843 - 1.621 - 1.359 - 1.070 -0.880 -0.761 - 0648 -0.538 -0.380
0.581 0.714 0.812 0.906 0.942 0.953 0.968 0.983 0.995
- 2.036 - 1.753 - 1.474 - 1.093 -0.895 -0.817 -0.702 -0.551 -0403
0.389 0.661 0.790 0,886 0.941 0.956 0.965 0.980 0.994
(Y/A> -2.214 -1.794 -1.458 -1.129 -0.892 -0.758 -0.691 -0.558 -0.392
WV
(Yl&
0.273 0.558 0.734 0.861 0.920 0.936 0.958 0.972 0.982 0.990
-2.187 -1.859 -1,524 - 1.151 -0.914 -0.832 - 0.695 -0.584 -0.501 -0.416
Table 2. Theoretical edge of moving region for rl = 35.5” Hopper included half-angle 5.81”
12.52
19.07”
@I~)
(Yl&
(xl&
(Yl&
0.879 0.895 0.917 0.943 0.964 0.983 0.994 0.999
- 2.487 -2.362 -2.176 - 1.894 - 1,630 -1.226 -0.998 -0.809
0.793 0.821 0.841 0.883 0.910 0.930 0.967 0.982 0.992 0.999
-2.321 -2,193 - 2.096 - 1.852 -1.667 -1.510 - 1.141 -0.943 -0.769 -0.612
(Y/x,
(xl&
(Y/x>
(xl&
-2.511 -2.306 -2.023 - 1,794 -1.515 -1.204 - 1.003 -0.879 - 0.763 - 0.599
0.467 0.653 0.754 0.840 0.918 0.946 0.952 0.967 0.919 0.996
-2.558 -2.193 - 1.923 - 1.618 - 1.236 - 1.059 -0.972 -0.888 -0.764 - 0.529
0.208 0.512 0.683 0.806 0898 0.936 0.951 0.968 0.982 0.994
W) 0.634 0.709 0.793 0.847 O+OO 0.943 0.966 0.916 0.986 0.996
54.5”
31.8’
265
G. C. GARDNER
Table 3. Theoretical edge of moving region fbr q = 40” Hopper included half-angle 660”
14.82
(xl&
(Y/4
0.892 0909 0.935 0.948 0972 0.985 0.994 0.999
- 2.494 -2.349 -2.106 - 1.934 - 1.551 - 1.281 - 1.072 -0.898
(-4) 0.684 0.749 0.798 0.856 0.886 0.925 0.960 0.976 0.987 0.998
25.1”
(Y/x) -2.813 -2.585 -2384 - 2.093 -1.918 - 1.632 - 1.294 -1.107 -0.941 -0.724
(xl&
(Y/a
0.517 0.653 0.750 0.816 0.880 0.932 0.961 0.912 0.984 0.996
50.0
35.6
-2.888 - 2.567 - 2.273 -2.021 -1.719 - 1.381 - 1.140 -1.028 -0.870 -0.631
m
(Y/a
0.292 0532 0666 0.771 0.852 0918 0.963 0.976 0.986 0.994
-3.100 -2.715 -2m4 - 2.086 -1.769 -1404 - 1.107 -0.932 -0.803 -0641
w 0.154 0442 0.614 0.744 0.828 0.910 0.944 0.954 0.964 0.972 0.982 0.995
(Y/h) -3.165 -2.808 -2.471 -2.125 -1.822 -1.418 -1.200 -1.115 -1.031 -0.949 -0.830 - 0642
Table 4. Theoretical edge of moving region for q = 50” half-angle
Hopper included 9.05”
40
16.3”
(xl&
(Y/a
0.733 0.860 0.890 0+08 0.922 0.952 0.982 0.990 0.996
-3.648 -2.979 -2.743 -2.577 -2QI40 - 2.058 - 1.628 - 1.355 -1.193
(xl& 0.573 0718 0.785 0.828 0.888 0.953 0.972 0.979 0.989 0.996 0.999
the wall boundary BC in terms of X = (x/A) and Y = b/A) where 1 is the half-width of the bin section. Different wall boundaries are given for different included hopper half-angles. It is noticed that the cohesion does not occur as a parameter in the results but the results are equally applicable to materials with cohesion as without it. The results for an angle of internal friction of 35.5” are drawn in Fig. 4 to illustrate that the shape of the curved boundary BC becomes progressively less curved as the included hopper angle becomes smaller.
(Y/3
c4v
-3.791 - 3.270 -2.974 -2.731 -2.349 - 1.746 - 1.506 - 1.396 - 1.200 - 1.029 -0.952
0.195 0.267 0460 0,665 0.727 0.801 0.877 0,931 0,962 0.973 0.982 0.997
-4.217 -4.112 - 3.732 -3.169 -2.944 - 2.626 -2.211 - 1.800 - 1.495 - 1.352 - 1.218 -0.923
THE EXPERIMENT
The theory has been concerned with a plane problem which, it has been explained, can be studied experimentally if the material tends to diverge very slightly rather than to converge in planes normal to the plane of interest and if the containing walls parallel to the plane of observation are smooth. Both these requirements are met in the apparatus. The theory has also been concerned with a stress system in which one family of slip-lines is tangent to the region of flow or the material fails within
266
The region of flow when discharging granular materials from bin-hopper X/X
Hopper
FIG. 4.
Theoretical
included
shape of edge of moving region. 7) = 35.5”.
itself at the boundary. This situation is attainable experimentally if the walls normal to the plane of observation are rough so that the shear stress, which would give slip of the material against the wall, is greater than for failure within the material itself. To make the walls rough a layer of the granular material was glued to them. This certainly makes the wall-material failure condition stronger than the material-material condition since the ability of the particles to rotate has been removed. There is, however, quite likely a transition layer adjacent to the walls in which the material strength properties change from one condition to the other and which may cause a difference between theory and practice. Such changes in material properties may indeed occur wherever there is a gradient of velocity normal to the direction of flow. The experimental
apparatus
The apparatus is illustrated in Fig. 5. The binhopper profile was provided by two vertical wooden walls, bevelled at the bottom so that two shorter plane walls could be fitted at any angle from 0” to
systems
65” to the horizontal immediately beneath them. The two sloping walls formed the hopper and their position could be adjusted to give a sharp edged slot discharge of any chosen width. The other two walls of the container were of Perspex, which is sufficiently smooth and allowed observation of the material. These walls were not quite vertical since the two vertical wooden walls, to which they were screwed, changed in width over their 4 ft height from 3 in. at the top to 3* in. at the bottom, thus avoiding any tendency to convergent flow normal to the plane of observation. The span between the wooden walls of the bin was 6 in. The internal wooden surfaces of the apparatus were coated with impact adhesive followed by a layer of the experimental granular material. The rough surface so formed proved durable for the number of experiments carried out. Discharge through the slot at the bottom of the hopper could be prevented by means of a hinged trap set about $ in. below the slot and held in place by an easily removed pin. When the pin was pulled out the trap swung free of the slot. The trap could also be rapidly swung back into place and refastened without disturbing the material in the bin-hopper system. The experimental material was an angular calcined flint named “Durite” of from & in. to & in. in size. It did not change shape or dust after passing it many times through the apparatus. It was also porous and thus easily dyed a deep blue by ink. It was determined, by means of a shear cell, that the “Durite” was cohesionless and had an angle of internal friction of 35.5”. Experimental
method
The system was filled with horizontal layers of “Durite” as shown in Fig. 5. The layer of undyed “Durite” was poured in and gently levelled with a stick. Then dyed material was poured against the Perspex wall and not over the whole cross section to limit the quantity used. The dyed layers were spaced more closely at the bottom of the apparatus than at the top. Material was discharged from the slot until the top surface in the bin was about 23 ft above the slot. 267
G. C. GARDNER
Wooden
side walls
/ Perspex front and back walls
Dyed
layers
et f
Hopper section in horizontal position
/ Swinqinq trap stop and start Dircharqe’slot
FIG. 5.
Experimental
268
apparatus.
The region of flow when discharging granular materials from bin-hopper
The slot width was about t in. which gave a convenient discharge rate of 5 cm3/sec and variation of the rate did not alter the experimental result. The bin was topped up again, without layering, and more material was discharged. The procedure was continued until the flow pattern was fully developed and the remaining outer edges of the horizontal dyed lines suffered no further erosion. Three methods were then used to determine the boundary of the moving material: 1. The position of the edge of the dyed layers nearest the axis of symmetry was measured in terms of x and y. Thus from one run four sets of measurements were obtained, two from each Perspex face. Each set was plotted individually together with the position of the edge of the slot and a smooth curve was drawn and extrapolated the small distance to the axis of symmetry. The points were then replotted using the intersection with the axis of symmetry as the origin and reducing the co-ordinates to (x/A) and (V/A). Only then were the results compared with the theoretical curve. The data obtained in this manner are given in Figs. 6 and 7. 2. Photographic time exposures of 3 set to 1 min duration were taken of the flowing material. The defined flow region was identical to that given by method 1 as is illustrated by Fig. 13. 3. A 6 in. depth of dyed material was poured into the top of the bin and run right through the apparatus while keeping the bin topped up with undyed material. It left a trail of blue material, which is clearly shown in Fig. 12 and which ran the whole height of the binhopper system. The trail lay inside the edge of the flow region defined by the horizontal blue layers, and, although it would all have been removed if more material had been allowed to flow through the apparatus, it nonetheless gave a clear definition of the fastest moving flow region. Measurements were taken from the outside edge of the trail and treated in the same manner as for method 1. The final results, in reduced co-ordinates, are plotted in Figs. 8 and 9. 269
l
systems
o m A
Position of the ends sets of dyed layers
X
Edqe of
Theoretical curve
discharqe
of
four
slot
‘I = 4o” ‘1 = 35*s”,
FIG. 6.
Shape of edge of whole flow region determined from horizontal dyed layers for hopper with horizontal Walls.
G. C. GARDNER
DOrn0
Position sets of
X
Edqe
of
of the ends of dyed layers
discharqe
four
I
slot
l
I% -
I
o D 6
x
Measurements trails
left
Edqe
of
from
four
by iluq discharqe
slot
C C
2-
I Q
n l
I-
h
7 .
m O! / =
.i
0 8
4 /
b
< i C1
Shape of edge of whole flow region determined from horizontal dyed layers for hopper with walls at 58” to the horizontal.
‘
x
FIG. 7.
FIG. 8. Shape of edge of fast moving region given by slug of blue material in hopper with horizontal walls.
270
The region of flow when discharging granular materials from bin-hopper
l 0
m 4
X
Measurements from trails left by $1~19 Edqe
of
discharqe
Experiments were carried out for hopper wall angles of 0” and 58” to the horizontal. Theoretically, the results should not vary for hopper angles from 0” to 355” to the horizontal, where 35.5” is the angle of internal friction.
four
Experimental observations slot
I 29
Shape of edge of fast moving region given by 9. slug of blue material in hopper with walls at 58” to horizontal.
FIG.
systems
Before the results are compared with theory, some experimental observations are important. Figure 10 shows the system shortly after flow had started. The flow region is a long ellipsoidal shape located on the axis of symmetry. As the flow region develops, this ellipsoidal region broadens and penetrates further up the bin section until the top free surface is reached. The flow region then continues to broaden but the sides, well above the discharge slot, become vertical and parallel. Measurementsaccordingtomethodl thengiveresults identical to those shown in Figs. 6 and 7 for the fully developed flow. The curve in the real x, y coordinates instead of the reduced co-ordinates is, of course, smaller by the difference in the scale factor 1. If only the bottom half of the bin is layered and undyed material is poured into the top half, the flow develops in the same manner except that, when the flow region reaches the unlayered material, it spreads rapidly. It appears that, even though the material was only very gently spread level by the stick in layering, a structure has been built into the layered material which makes it stronger than material which either has recently flowed or is in a state of flow. The flow in the layered material spreads very slowly to cover the whole possible flow region. When the fully or partly developed flow is stopped, as shown in Figs. 10-12, and then started again, the flow region continues to develop from the point at which it stopped, indicating that the layering operation is the cause of the observed development of flow. The slow spread of the flow region illustrates what may happen in bunkers that have been closed down for a period. It also introduces an element of doubt into these experiments that have previously been performed with layered materials and have been concerned only with initial deformations. An example of this is JOHANSON’S [7] experiments to co&m the JENIKEand SHIELD[6] flow theory. KVAPIL [8] also developed a theory of bin-hopper 271
G. C. GARDNER
design in which he observed the elongated ellipsoidal shape and used its dimensions as design data. Another observation of importance is that just above the discharge slot there is a tongue of material moving much faster than the surrounding material. This has been observed by others, for instance BROWN and HAWKSLEY [2]. The tongue is seen in some of the time exposures but most clearly in Fig. 11 where the flow was stopped when the blue plug of material was just beginning to discharge. The blue tongue moved rapidly ahead of the rest of the blue material. Lastly it is noted that the edge of the flow region never reached the bin walls completely. The walls were sufficiently rough to ensure that the material failed within itself and not with respect to the wall. Discussion of the experimental results Figures 6 and 7 show the experimental results obtained by method 1 where the position of the ends of the dyed layers was measured. This defines the complete flow region. The theoretical curves for angles of internal friction 30”, 35.5” and 40” are also plotted on Fig. 6 for the hopper with a wall angle to the horizontal of 0”. The theoretical line for the measured angle of internal friction 355” fits the data best and indeed fits it very well. It is, however, noticeable that the experimental points tend to cross the theoretical line. At the base of the hopper they correspond better with the theoretical line corresponding to an angle of internal friction of 40” and near the bin section with the theoretical line corresponding to an angle of internal friction of 30”. It has been indicated that differences between theory and experiment may be due to the choice of the stress system, the variation in the strength properties of the material, especially near the boundary of the flow region, and lastly due to neglect of kinetic stresses, especially near the discharge slot where the fast moving tongue of material occurs. Figure 7 is for the hopper wall at an angle of 58” to the horizontal. The same comments apply but the correlation is equally good. Figures 8 and 9 show the results obtained by method 3 from the trail of the plug of dyed material. Where the angle of the hopper wall to the horizontal is 0”, the correlation appears to be quite good near
the bin section but diverges considerably from theory near the discharge slot. The discrepancy is even greater when the angle of the hopper wall to the horizontal is 58”. In both cases the edge of the fast-moving core ilow, defined by this experimental method, appears to be a straight line through the origin for the greater part of the hopper flow region. It must be concluded that the theory does not give an adequate description of the measurements by this method. It should be noted that the extent of the fastmoving core flow is probably of importance in problems concerning the rate of discharge of materials and should be used in say the theory of BROWN [l]. On the other hand the shape of the whole flow region will be important in the design of bin-hopper systems to ensure that there are no “dead” spaces which could prove troublesome with respect to cohesive materials. It is re-emphasised that the theory is applicable to cohesive material but has only been tested experimentally for a non-cohesive material. Experiments with cohesive materials are desirable and might determine whether the angle of internal friction or JENIKE’S[5] apparent angle of internal friction are important. CONCLUSIONS
A theory has been developed, equally applicable to cohesive and non-cohesive materials, which gives the shape of the flowing region in the transition from a bin section with vertical parallel walls to a hopper of converging plane walls. The results are given in Tables l-4 and in terms of reduced co-ordinates (x/A) and b/A) where 1 is the half span of the bin section. The theory was tested for an angular non-cohesive material with an angle of internal friction of 35.5”. The comparison of theory with experiment is good but it is noted that there are consistent differences which, although they may be of negligible importance in defining the transitional wall shape, may be important in other problems concerning the flow of granular materials such as the rate of discharge through the discharge slot. Near the outlet slot the difference may be attributed to the influence of a tongue of fast-moving material projecting a small 272
The region of flow when discharging granular materials from bin-hopper
way up into the mass. This tongue is well known under free discharge conditions. Near the bin section the difference may be due either to the change of properties of the material between the fast-moving
core
and
the flow boundary
or to the
of particular stress fields in the theory. These stress fields are fully discussed. The experiment also showed that the shape of the fast moving core could be well defined by allowing a plug of dyed material to pass through the apparatus.? It left a trail at its edge which lay well within the boundaries of the complete flow region. Results obtained by this method are presented.
systems
He also devised the method of deiining the extent of the fast moving region using the blue slug of material. Mr. J. C. RICHARDSof B.C.U.R.A. provided the calcined
flint and measuredits ande of internal friction.
The work was carried o;t at the Central Electricity Research Laboratories and the paper is published by permission of the Central Electricity Generating Board.
assumption
AcknowledgementsMr. D. LATHAMcarried out the experimental part of the work during his undergraduate vacation.
7 It has recently come to the writer’s notice that J. B. MARTIN and J. C. RICHARDS(J. Science Technology 1965 11 31) found this method convenient with respect to conical hoppers.
NOTATION Cohesion Twice the angle between the x direction and the g direction of action of the major principal stress n Constant exponent P Stress parameter defined by Fig. 2 PO Constant value of stress parameter r Radial co-ordinate X Horizontal co-ordinate Vertical co-ordinate Material density r’ Stress characteristic parameter defined by Eq. (18) & Angle of internal friction Angular co-ordinate 3 Half-width of flow region in the bin h (n/4 - 77/2) P u Normal compressive stress Shear stress Stress characteristic parameter defined by ,Eq. (17) i C
REFERENCES
111 BROWNR. L., Nature, Lond. 1961 191458. PI BROWNR. L. and HAWKSLEY P. G. W., Fuel, Lond. 1947 26 159. DELAPLAINEJ. D. A.1.ChX.X
1956 2 127.
;:; GARDNERG. C., Chem. Engng Sci. 1964 19 283. PI JENIKEA. W., Bulletin No. 108 of Utah Engineering Experiment Station 1961. 161 JEN~E A. W. and SHIELDR. T., J. appl. Mech. 1959 81B 599. t71 181 t91 1101
C
JOHANSONJ. R., J. appl. Mech. 1964 31 499. KVAPIL R., Theorie der Schuttgutbewegung V.E.B. Verlag Technik, Berlin, 1959. O’CALLAGHANJ. R., J. agric. Engng Res. 1960 5 200. SOKOLOVSKI V. V., Statics of Soil Media pp. 14 and 139, Butterworths, 1960.
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