DETERMINING FLOW PROPERTIES OF PARTICULATE MATERIALS

DETERMINATION DES CARACTERISTIQUES D'ECOULEMENT DE MATERIAUX PULVERULENTS

English Abstract

DETERMINING FLOW PROPERTIES
OF PARTICULATE MATERIALS
ABSTRACT
Apparatus and a test method for bench scale
determination of whether a particulate material will
flow under the action of gravity alone from an outlet
in the bottom of a container. The apparatus includes a
test cell (12) having inclined conical side walls (14),
and that is closed at its larger end by a plug (10)
having an inwardly-facing surface (18) that is concave.
In this way, cylindrical surfaces are avoided, and the
shape of the space within the test cell offers minimal
interference with the plastic stress field of the
material. The testing method includes the novel step
of inverting the test cell after consolidation of the
test material but prior to application of the failure
load.

Claims

-19-
THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE PROPERTY OR
PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:
1. A method of bench scale testing to ascertain
a flow-related property of a particulate material,
comprising the steps of:
a) providing a hollow test cell that includes a
conical section defined by an inwardly-facing surface con-
forming to a section of height C of a cone, the conical
section having a smaller-diameter end and a larger-diameter
end of diameter D, and that further includes a plug located
adjacent to and spanning the larger-diameter end of the
conical section, but slightly smaller in diameter than it,
so as to be freely movable axially a limited distance into
the space within the conical section, the plug having a
concave inwardly-facing surface;
b) placing the test cell on a supporting surface
with the larger-diameter end of the conical section below
its smaller-diameter end and with the plug supported in
its aforementioned location adjacent the larger-diameter
end of the conical section;
c) filling the test cell completely full with
the particulate material;
d) consolidating the particulate material within
the test cell by applying a downward pressure at the
smaller-diameter end of the conical section, so that the
direction of motion of the particulate material during
consolidation is toward the plug;
e) inverting the test cell with the plug and the
consolidated particulate material still in it, so that
after the inversion the larger-diameter end of the conical
section is above its smaller-diameter end;
f) applying a gradually-increasing downward load
to the plug to produce failure motion of the particulate
material toward the smaller-diameter end of the conical
section; and,
g) noting the downward load at which failure
occurs.

-20-
2. The method of Claim 1 wherein the flow-related
property is the minimum diameter Rmin of an outlet in
the bottom of a full-size container at which flow of the
particulate material will occur by gravity alone; and
wherein the step of consolidating the particulate material
further includes the step of applying a pres-
sure to the particulate material at the smaller-diameter
end of the conical section equal to the consolidation
pressure at the corresponding location in the full-size
container; and wherein, following the step of noting the
downward load, the method is further characterized by the
step of calculating the minimum diameter at which flow
will occur by the equation
Bmin= <IMG> where
LF is the noted downward load,
A1 is the average cross sectional area of the test
cell, and
?1 is the bulk density of the particulate material.
3. The method of Claim 2 wherein the particulate
material has been at rest in the full-size container for
an appreciable time, and wherein the step of consolidating
further includes applying the consolidation
pressure for an interval of time equal to how long the
particulate material in the full-size container has been
at rest.
4. The method of Claim 1 wherein the flow-related
property is whether or not the particulate material will
flow, after having been at rest for some time, from a
full-size container that requires gravity flow for

-21-
discharge and that has an outlet of diameter B in its
bottom; and wherein the step of consolidating the
particulate material within the test cell by applying a
downward pressure further includes the step
of applying a consolidation load Lc uniformly distri-
buted over the particulate material at the smaller-dia-
meter end of the conical section, so that the direction of
motion of the particulate material during consolidation
is toward the plug, where
Lc= ? F?1A1B
where F = overpressure factor,
?1= bulk density of the particulate material
being tested,
A1= average cross sectional area of the test
cell;
and wherein the method o f Claim 1 further includes
the step, following the step of noting the downward
load, of determining whether the noted load at which
failure occurred is less than
Lf= <IMG>
whereby, if it is, then flow from the full-size container
will occur by gravity, but if it is not, then flow from
the full size container will not occur by gravity.

//
-22-
5. The method of Claim 4 wherein the consolida-
tion load Lc is applied to the particulate material
for an interval of time equal to how long the particulate
material in the full-size container has been at rest.
6. The method of Claim 1 wherein the flow-related
property is whether or not the particulate material will
flow, after having been at rest for some time, from a
full-size container that requires gravity flow for dis-
charge and that has an outlet of diameter B in its
bottom; and before the step of filling the test cell,
further including the step of providing a mold
ring that is hollow and that has a cylindrical inside
surface of diameter equal to the diameter of the smaller-
diameter end of the conical section; and before the step
of filling the test cell, further including the
step of resting the mold ring on top of and coaxial with
the test cell to form an upward extension of it; and
after the step of filling the test cell, further
including the steps of filling the mold ring with the
particulate material and placing a consolidation load
disk having an outside diameter slightly less than the
inside diameter of the mold ring on top of the particulate
material in the mold ring; and wherein the step of con-
solidating the particulate material is accomplished
by applying a consolidation load Lc uniformly distributed
over the top of the particulate material in the mold ring
by the consolidation load disk, so that the direction of
motion of the particulate material during consolidation is
toward the plug, where
Lc = ? F?1A1B

-23-
and F = overpressure factor,
?1= bulk density of the particulate material being
tested,
A1= average cross sectional area of the test cell;
and wherein, before the step of inverting the test cell,
the method further includes the steps of con-
tinuing to apply the consolidation load for an interval
of time equal to how long the particulate material in
the full-size container has been at rest, and removing
the mold ring and the particulate material contained in
it from the test cell; and after the step of noting the
downward load, the method further includes
the step of determining whether the noted load at which
failure occurred is less than
Lf = <IMG>
whereby, if it is, then flow from the full-size container
will occur by gravity, but if it is not, then flow from
the full size container will not occur by gravity.
7. The method of Claim 1 wherein the flow-related
property is the unconfined yield strength of the parti-
culate material and wherein, in the consolidating step
the downward pressure .sigma.c is given by
.sigma.c = ? + ? ?1 C
w where Lc= consolidation load
A1= average cross sectional area of the test cell
?1= bulk density of the particulate material;

-24-
and, after the step of noting the load, the method
further includes the step of calculating the
unconfined yield strength by the equation
fc = <IMG>
where h = 2.1.

-25-
8. Apparatus for use in ascertaining the flow
properties of a particulate material, comprising:
a hollow test cell that includes a conical section
defined by an inwardly-facing surface conforming to a
section of a cone, said conical section having a smaller-
diameter end and a larger-diameter end; and,
a plug located adjacent to and spanning the larger-
diameter end of said conical section; but slightly smaller
in diameter than it, so as to be freely movable axially
a limited distance into the space within said conical
section, said plug having a concave inwardly-facing
surface.
9. The apparatus of Claim 8 wherein the vertex
semi-angle of the conical section is in the range between
a minimum of 4 degrees and a maximum of .theta.c , where .theta.c
is the largest angle compatable with the mass-flow stress
field in the test cell.

-26-
10. A method of bench scale testing to ascertain
a flow-related property of a particulate material,
comprising the steps of:
a) providing a hollow test cell that includes
a frustrated wedge section defined by an inwardly-facing
surface conforming to a section of height C of a wedge,
the frustrated wedge section having a smaller end and a
larger end of width D, and that further includes a plug
located adjacent to and spanning the larger end of the
frustrated wedge section; but slightly narrower than it,
so as to be freely movable a limited distance into the
space within the frustrated wedge section; the plug having
a concave trough-like inwardly-facing surface;
b) placing the test cell on a supporting surface
with the larger end of the frustrated wedge section below
its smaller end and with the plug supported in its
aforementioned location adjacent the larger end of the
frustrated wedge section;
c) filling the test cell completely full with
particulate material;
d) consolidating the particulate material with-
in the test cell by applying a downward pressure at the
smaller end of the frustrated wedge section, so that the
direction of motion of the particulate material during
consolidation is toward the plug;
e) inverting the test cell with the plug and the
consolidated particulate material still in it, so that
after the inversion the larger end of the frustrated wedge
section is above its smaller end;
f) applying a gradually-increasing downward load
to the plug to produce failure motion of the particulate
material toward the smaller end of the frustrated wedge
section; and,
g) noting the downward load at which failure
occurs.

-27-
11. The method of Claim 10 wherein the flow-
related property is the minimum width BMIN of an outlet
in the bottom of a full-size container at which flow
of the particulate material will occur by gravity alone;
and wherein the step of consolidating the particulate
material further includes the step of apply-
ing a pressure to the particulate material at the smaller-
diameter end of the conical section equal to the con-
solidation pressure at the corresponding location in the
full-size container; and wherein, following the step of
noting the downward load, the method further includes
the step of calculating the minimum width at
which flow will occur by the equation
BMIN = <IMG> where
Lf is the noted downward load,
A1 is the average cross sectional area of the test
cell, and
?1 the bulk density of the particulate material.
12. The method of Claim 11 wherein the particulate
material has been at rest in the full-size container for
an appreciable time, and wherein the step of consolidating
further includes applying the consolidation
pressure for an internal of time equal to how long the
particulate material in the full-size container has
been at rest.
13. The method of Claim 10 wherein the flow-
related property is whether or not the particulate
material will flow, after having been at rest for some time,
from a full-size container that requires gravity flow for
discharge and that has an outlet of width B in its

-28-
bottom; and wherein the step of consolidating the parti-
culate material within the test cell by applying a down-
ward pressure further includes the step of
applying a consolidation load Lc uniformaly distributed
over the particulate material at the smaller end of the
frustrated wedge section, so that the direction of motion
of the particulate material during consolidation is toward
the plug, where
Lc= F?1A1B
where F = overpressure factor,
Y, = bulk density of the particulate material
being tested,
?1 = average cross sectional area of the test
cell;
and wherein the method of Claim 11 further
includes the step, following the step of noting the
downward load, of determining whether the noted load
at which failure occurred is less than
<IMG>
whereby, if it is, then flow from the full-size container
will occur by gravity, but if it is not, then flow from
the full-size container will not occur by gravity.
14. The method of Claim 13 wherein the consolida-
tion load Lc is applied to the particulate material for
an interval of time equal to how long the particulate
material in the full-size container has been at rest.

-29-
15. The method of Claim 10 wherein the flow-
related property is whether or not the particulate material
will flow, after having been at rest for some time, from
a full-size container that requires gravity flow for
discharge and that has an outlet of width B in its
bottom; and before the step of filling the test cell,
further including the step of providing a mold
frame that is hollow and that has a rectangular inside
surface of dimensions equal to those of the smaller end
of the frustrated wedge section: and, before the step of
filling the test cell, further including the step
of resting the mold frame on top of and registered with
the test cell to form an upward extension of it; and,
after the step of filling the test cell, further chara-
terized by the steps of filling the mold ring with the
particulate material and placing a consolidation load
plate having outside dimensions slightly less than the
inside dimensions of the mold frame on top of the particu-
late material in the mold frame; and wherein the step of
consolidating the particulate material is accomplished
by applying a consolidation load Lc uniformly distributed
over the top of the particulate material in the mold frame
by the consolidation load plate, so that the direction of
motion of the particulate material during consolidation
is toward the plug, where
Lc = F?1A1B
and F = overpressure factor,
?1= bulk density of the particulate material
being tested,
and A1= average cross sectional area of the test cell;
and wherein, before the step of inverting the test cell,
the method further including the steps of
continuing to apply the consolidation load for an interval
of time equal to how long the particulate material in the
full-size container has been at rest, and removing the

-30-
mold frame and the particulate material contained in it
from the test cell; and, after the step of noting the
downward load, the method further including
the step of determining whether the noted load at which
failure occurred is less than
<IMG>
whereby, if it is, then flow from the full-size container
will occur by gravity, but if it is not, then flow from
the full-size container will not occur by gravity.
16. The method of Claim 10 wherein the flow-
related property is the unconfined yield strength of
the particulate material and wherein, in the consolidat-
ing step the downward pressure .sigma.c is given by
<IMG>
where Lc = consolidation load
A1 = average cross sectional area of the test cell
and ?1 = bulk density of the particulate material;
and, after the step of noting the load, the method
further includes the step of calculating the
unconfined yield strength by the equation
<IMG>
where h = 1.1.

-31-
17. Apparatus for use in ascertaining the flow
properties of a particulate material, comprising:
a hollow test cell that includes a frustrated
wedge section defined by an inwardly-facing surface con-
forming to a section of a wedge, said frustrated wedge
section having a smaller end and a larger end; and,
a plug located adjacent to and spanning the larger
end of the frustrated wedge section; but slightly
narrower than it, so as to be freely movable a limited
distance into the space within the frustrated wedge sec-
tion; said plug having a concave trough-like inwardly-
facing surface.
18. The apparatus of Claim 17 wherein the vertex
semi-angle of the frustrated wedge section is in the
range between a minimum of 4 degrees and a maximum of
.theta.p , where .theta.p is the largest angle compatable with
the mass-flow stress field in the test cell.

Description

--1--
INTERNATIONAL APPLICATION
UNDER THE
PATENT COOPERATION TREATY
DESCRIPTION
DETERMINING FLOW PROPERTIES
OF PARTICULATE MATERIALS
Inventors: JOHANSON, Jerry Ray and
JOHANSON, Kerry Dee
Technical Field
The present invention is in the field of bulk
particulate solids and more specifically relates to an
apparatus and testing method for determining on the
basis of bench scale testing whether particulate matter
will flow under the action of gravity through an outlet
in the bottom of a container.
Background Art
Bulk solids in a divided state such as flour, sugar,
ores, dry chemicals and coal are generally handled in
silos or containers that require gravity flow for dis-
charge. One of the problems of designing such containers
is sizing of the outlet so that the solids do not form an
obstruction by arching across the outlet.
The size of outlet required to prevent arching is a
function of the unconfined yield strength of the material,
the density of the material, and the shape of the outlet.
There are basically two methods for determining this
critical dimension. First, one could construct a full
'~
.. "; .. . . . . .

z
--2--
size hopper, load it with the material, and observe
whether flow takes place from the outlet. The other
approach known in the art is to measure the unconfined
yield strength, and then to use this unconfined yield
strength as a function of consolidation pressure to
analyze the results to predict the ~opper outlet dimen-
sion. This latter approach has been described i~ the
following technical papers: Jenike, A. W., P. J. Elsey,
and R. H. Woolley. "Flow Properties of Bulk Solids."
Proc. Amer. Soc. Test. Mater. 60:1168-1181, 1960;
Jenike, A. W. "Gravity Flow of Bulk Solids." Univ. of
Utah Engineering Experiment Station Bulletin No. 108,
1961; Johanson, J. R. "Know Your Material--How to Predict
and Use the Properties of Bulk Solids. n Chem. Eng.,
October 30, 1978, pp. 9-17.
This latter method is usually accomplished by means
of a direct shear tester and therefore it is rather
cumbersome and time consuming to obtain the results. A
quicker and easier method is needed, particularly for use
in the field, where portable testing equipment would be
especially handy.
Disclosure of the Invention
It is an object of the present invention to provide
apparatus and a bench scale test method to determine
whether flow will occur through a given outlet under the
action of gravity alone.
It is a further object to provide apparatus and a
test method for determining the unconfined yield strength
of a particulate material.
A further objective is to provide an apparatus and
test method that is applicable to conical containers as
well as t~ containers of rectangular horizontal cross
section.
These objectiveq are accomplished by a novel test
chamber. The portion of the test chamber in which the

--3--
material is tested includes no cylindrical walls t~lat
contact the material, and in this way the uncertainty
associated with the frictional effects of cylindrical
walls is eliminated. The side walls of the test chamber
conform to the surface of a cone, and the larger end of
the conical section is closed by a plug that includes an
inwardly facing coaxial concave surface. This unique
shape of the test chamber affords minimum interference
with the plastic stress field present in the material.
During the consolidation phase of the test, the conical
walls provide an increasing cross sectional area in the
direction of motion so as to minimize the frictional
effects of the walls. During the failure phase of the
test, the same conical walls provide a decreasing cross
sectional area in the direction of motion to insure an
arching condition similar to that occurring in a hopper
during failure conditions.
In a novel aspect of the testing method, the
material to be tested is consolidated in the test cell
with the concave plug at the bottom of the test cell,
and then the entire test cell including the concave plug
is turned upside down for the failure loading portion
of the test.
These and other objectives and advantages of the
present invention will become clear from the detailed
description given below in which several preferred
embodiments are described in relation to the drawings.
The detailed description is presented to illustrate the
present invention, but is not intended to limit it.
Brief Description of the Drawings
Figure 1 is an elevational view in cross section
showing the apparatus of a preferred embodiment of the
present invention at the initial consolidation phase of
the test procedure;

3)Z
~4~
Figuxe 2 is an elevational view in cross section
showing the apparatus at a later stage of the test
procedure;
Figure 3 is an elevational view in cross section
showing the apparatus at a later stage of the test
procedure;
Figure 4 is a perspective view showing a rectangular
test apparatus;
Figure 5 is an elevational view in cross section
showing the apparatus of a second preferred embodiment of
the present invention at the initial consolidation ph~se
of the test procedure; and,
Figure 6 is an elevational view in cross section
showing the apparatus of Figure 5 at the start of the
failure loading phase of the test procedure.
Best Mode for Carrying Out the Invention
In the first part of this section, the structure
of the apparatus will be described in detail. Next, the
methods of using the apparatus in carrying out various
tests will be discussed. Finally, several alternative
embodiments and variations will be described.
Turning now to the drawings in which like parts are
denoted by the same reference numeral throughout, there
is shown in Figure 1 a preferred embodiment of the
apparatus that is used for performing the tests described
below.
The apparatus includes a consolidation base 20 which
rests ~n the floor or on a bench. The conical plug 10
rests on the base 20 with its conical surface 18 facing
the inside of the test cell 12. The shape of the space
enclosed by the test cell 12 and by the conical plug 10
is an important aspect of the present invention, as will
be described below.

--5--
The test cell 12 is a unitary sleeve-like part
having a conical inside surface 14 shaped to conform to a
truncated cone joined at its larger diameter to a
cylindrical inside surface 16. The conical inside
surface 1~ is inclined to the vertical by appro~imately
six degrees, so that it could also be said that the cone
to which the surface 14 conforms has a vertex angle of
twelve degrees, in the preferred embodiment. If the
semi-angle is less than four degrees, the convergence
will be insufficient to form an arch during the failure
step of the test procedure. On the other hand, the
angle must not be so great that it becomes incompatable
with the mass flow stress field in the test cell durinc~
the failure step of the test procedure. The conical
plug 10 is slightly smaller in diameter than the cylin-
drical inside surface 16 so that the plug 10 can, at a
later stage of the test, move freely in the axial
direction with respect to the test cell 12. The height
of the conical plug 10 in the axial direction is
slightly less than the height of the cylindrical inside
surface 16 to accommodate some movement by the plug in
the axial direction.
The apparatus further includes a mold ring 24 that
rests upon the test cell 12 and that has a diameter
approximately equal to the diameter of the opening at
the top 34 of the test cell.
In a typical test, the test cell 12 and the mold
ring 24 are filled with the material 32 under test, and
the material 32 is then consolidated to a specific
degree.
In the best mode for carryi~g out the invention,
the consolidation is accomplished by placing the flat
disk 26 on top of the material 32 under test. The

-6-
diameter of the flat disk is slightly less than the
inside diameter of the mold ring 24 to permit the disk 26
to move downwardly into the mold ring 24 without contact-
ing the mold ring. In the best mode, a consolidation
load container 28 is then rested upon the disk 26 and is
slowly filled with a particulate ballast material 30
to a specific pre-calculated depth H. All of the part:s
of the apparatus are made of steel in the preferred
embodiment, although in other embodiments any strong
durable non-absorbent material may be used.
Because the material in the full-size hopper
typically includes a degree of moisture, the material 3
under test also contains moisture, and it is important
to prevent the loss of this moisture, since it affects
the accuracy of the results. For this reason, a metal
or rubber keeper ring 22 tightly encircles the crack
between the test cell 12 and the base 20.
The length of time during which the consolidation
load is applied should approximate the dead time of the
material in the proposed hopper, and if this amounts to
an appreciable amount of time, it may be essential to
take the additional step against moisture loss shown in
Figure 2. In Figure 2, a moisture seal 40 is inserted
between the material 32 under test and the disk 26.
The moisture seal 40 consists of a thin sheet of
pliable plastic, which is held in place by the keeper 42.
The keeper 42 in a preferred embodiment consists of a
band of metal or rubber.
Figure 3 shows the apparatus at a later stage of
the test method. After the material being tested has
consolidated for the desired length of time, the moisture
seals are removed and the test cell 12 is turned upside
.,,,, ~ . .

_7_ h~
down and placed on the failure base 48. The failure
base 48 is merely a support which does not interfere
with the material in tXe test cell or prevent the
material 32 from falling out of the test cell. Note that
in Figure 3, the test cell 12 is upside down relative
to its position in Figures 1 and 2, and that the conical
plug 10 remains in place in the test cell. On top of the
conical plug is placed the failure disk 54 which, in a
preferred embodiment, is a disk of steèl which serves to
evenly distribute the applied failure load. The failure
load container 50, which is not necessarily the same a~
the consolidation load container 28, is placed on top
of the failure disk 54 and is gradually filled with a
ballast material, such as sand, gravel, or water. At
some point as additional failure load material 52 is added,
the pressure loading on the material 32 becomes too grea'
to resist, and the material 32 collapses and falls out oE
the test cell 12. The height Hl of the ailure load
material 52 at the instant of failure is carefully noted
and, as will be seen below, is used in calculating the
test results.
It will be recognized that the conical shape of
the test cell is an important aspect of the invention.
During the consolidation phase of testing, shown in
Figures 1 and 2, this shape provides an increase in area
in the direction of flow, insuring a minimum of fric-
tional reaction from the walls of the test cell during
the consolidation phase, thereby permitting a nearly
uniform and known compaction of the material tested.
During the failure phase of testing, shown in Figure 3,
the conical shape of the test cell provides a decreasing
cross sectional area in the direction of flow so as to
simulate the condition occurring in a full-scale hopper.
The apparatus shown in Figures 1-3 is used for
determining whether a particulate material in a large

-8- ~ 2
container will flow by the action of gravity alone
through a circular outlet in the bottom of the container.
If, instead of being circular, the outlet is rectangular,
the apparatus shown in Figure 4 may be used. That
apparatus is in every sense the rectangular analog of the
apparatus shown in Figures 1-3.
In the apparatus of Figure 4, a trough-shaped
plug 60 having the inclined plane faces 68 extends into
the test cell 62. The test cell 62 is rectangular in
horizontal cross-section. The right and left walls as
shown in Figure 4 each include a vertical portion 66 and
an inclined portion 64. The front and rear walls are
vertical only, and are provided with removable panels 84
tha' are held in place by removable pins of which the
pin 72 is typical.
Figure 4 is comparable to Figure 1 in that it shows
the apparatus in the initial stage of the testing. A
rectangular mold frame 74 rests on top of the test cell 62,
and is comparable to the mold ring 24 of Figure 1. A
consolidation plate 76 is interposed between the con-
solidation load container 78 and the material 82 under
test. The entire apparatus rests on a consolidation
base 70.
With reference to the apparatus of Figures 1-4
inclusive, it is essential when testing cohesive solids
(defined as those in which the ratio R of the compacting
pressure to the instantaneous unconfined yield stress is
less than 2.0) that the walls of the converging parts
of the test cell have a rough finish and that the height-
to-diameter ratio C/D should not exceed 0~2 R. If the
ratio exceeds 0.2 R, there is a considerable likelihood
that recompaction of the material will occur during
failure, thereby giving a false reading. Since the
ratio R is seldom less than 1.1 for computing pressures of
practical interest, in the preferred embodiment, C/D has

- 9 -
been chosen to equal 0.22. For the embodiment shown in
Figure 4, the critical value of C/D is about twice that
for the configurations of Figures 1-3, and in a pre-
ferred embodiment, the ratio C/D is approximately equal
5 to 0.4 R or 0.44 for R = 1.1.
This relatively larger allowable value of C for
the rectangular test cell can be very important when
testing solids with large particles where C must be
several times larger than the particle size.
The test procedure is substantially the same for the
embodiments of Figures 1 and 4. The apparatus shown
and described, when used in accordance with the test
procedure given below, can be used to predict whether a
circular or rectangular outlet in the bottom of a con-
tainer partially filled with a particulate material will
discharge under the action of gravity alone or whether,
instead, arching of the material above the outlet will
develop to prevent the desired flow. It will be seen
below, that in addition to determining whether or not
~low will occur, the method can be used to determine
the size of outlet required to prevent arching and, with
some minor modifications, can be used to measure the
unconfined yield strength of the material under test.
Initially, the test cell 12 with the conical
plug 10 are resting on the base 20. The mold ring 24 is
assumed to be resting on the test cell 12. The test
cell is filled with the material under test to the top
of the mold ring. Next, the disk 26 is placed on top
of the material 32, the consolidation load container 28
is placed on top of the disk 26, and the consolidation
load 30 is poured into the container 28. This causes
the material 32 to consolidate, and if the material 32
has consolidated to a level below the top 34 of the test
cell, it is necessary to remove the consolidation con-
.

-10-
tainer, to refill the mold ring 24, and to apply the
consolidation load 30 again to the material 32 until the
consolidation le~el does not sink below the top 34 of
the test cell.
Next, the mold ring 24 is removed and the
material 32 is scraped level with the top 34 of the test
cell. Thereafter, the disk 26 and the consolidation
load container 28 are replaced as shown in Figure 2, an~
the consolidation load 30 is applied and allowed to sit
for a time e~ual to the time at rest of the material
within the container being simulated. If this time at
rest is in excess of a few minutes, it is necessary to
cover the test cell with a moisture impermeable seal such
as the moisture seal 40 shown in Figure 2.
When the apparatus is used to test whether or not
flow will occur, the appropriate height H of ballast
in the consolidation load container 28 is:
F ~, A, \~/c
H = , ~,~, Y A B -- ~A
where
Y,= bulk density of material in test cell
~ = bulk density of ballast material
A,= average cross sectional area of test cell
A~= cross sectional area of consolidation
load container
Wc= weight of the consolidation load container
= diameter of circular outlet of a conical
con-tainer or width of rectangular outlet
of a rectangular container
~ = l for a conical container
~v = 0 for a rectangular container
F = overpressure factor
The overpressure factor ~ accounts for the addi-
tional pressure caused by forces other than gravity that
may be operative on the container being simulated.
:

4~
Examples include vibration, the impact of a falling
stream of particles entering the container, and the
presence of gas pressure gradients within the material
in the container such as might be caused by a flow of
gas through the solids from top to bottom.
If vibrations cause a peak vertical acceleration
of a, then
F= (1~ ~ )
where g is the acceleration of gravity.
A downwardly acting gas pressure gradient of
magnitude dp/dx can be accounted for by setting
F=(l+,ol~/d~)
where Y is the bulk density of the material in the
container and dp/dx is the pressure gradient in the
upward direction.
The appropriate expression for F in the presence
of a falling stream of particles is derived in "New
Design Criteria for Hoppers and Bins" by J. R. Johanson
and H. Colijn Iron & Steel Engineer, October 1964,
pp. 85-104.
The consolidation pressure is given by
6 - F ~' ~
,
and the consolidation load is
LC=F YA B = WC~Y~A2H
~fter the consolidation time has elapsed, the con-
solidation load container 28, the disk 26, and the
, .
,
.
.~ .

-12-
moisture seal 40 are removed and the test cell 12 along
with the base 20, and including the conical plug 10 are
turned upside down and placed on the failure base 48 in
the position shown in Figure 3. Assuming the material 32
has a measurable cohesion, it should not fall out of
the test cell in response to the shocks normally
encountered in handling the apparatus, although clearly,
shocks are to be avoided. Thereafter, the base 20 is
removed and the failure disk 54 and the failure load
container 50 are stacked on top of the conical plug 10
as shown in Figure 3. It is important that the disk 54
and the container 50 do not touch the te~t cell during the
remaining phases of the test.
Next, the failure load 52 is very gradually poured
into the failure load container 50 until failure of the
material in the load cell occurs. Typically, such
failures are rather abrupt and most of the material w;th-
in the test cell will fall out upon failure. Very little
actual movement of the conical plug 10 occurs before
ailure, and this limited movement is permitted by the
clearance between the conical plug 10 and the inside of
the test cell. The height Hl of the material 52 wi~hin
the failure load container 50 at the time of failure is
determined by leveling the material in the failure load
container 50 and measuring the height Hl shown in
Figure 3.
If this value of height Hl satisfies the following
inequality, then flow will occur by gravity from the
outlet ^f diameter B;
~ C A Y )

-13- 1~9~49~
- where C is the height o the test cell and D is the
diameter of the test cell, as shown in Figure 1.
As an alternative to increasing the value of ~1~
the force LF applied to cause failure could be measured
and the critical failure condition could be described by
the equation:
LF ~ ( D ~ I)C A,Y,
The test procedure for the rectangular opening
using the apparatus of Figure 4 is identical to the pro-
cedure iust described with the exception that the valueof m is different from that used for the circuiar openin~,
and the symbol B is defined to be the width of the
rectangular outlet which has vertical inlet walls or is
infinitely long, and D is the width of the rectangular
test cell. It is seen that these differences affect on]y
the magnitude of the consolidation load and the calcu-
lated value of Hl, but do not otherwise affect the steps
of the test procedure.
Sometimes it is desirable to measure the unconfined
yield strength of the material, and this can be done
using a relatively minor variation on the above test
procedure. The only differences where the unconfined
yield strength is to be determined are that the applied
consolidation pressure should equal
2S ~ = LC + y C
where the symbols have the same meaning as above, and
if the applied load at failure is denoted by LF then the
unconfined yield strength fc may be calculated by use of
,
~. , .
..... ,.. , - ~ - - :
'
~;:

-14- ~ 9Z
the equation
~c - h (~' A, C
where h is 2.1 for the conical test cell and h is 1.1
for the rectanaular test cell.
In another minor variation on the method for
determining whether bulk solids will or will not flow
from an outlet in a container, it is recognized that
there are other techniques for applying a consolidation
pressure to the material 32 in the test cell. For
example, a pneumatic or hydraulic ram could be used, or
even a screw jack could be used with a load cell to
indicate the force applied. Regardless of the device
used to apply the consolidation pressure, the con-
solidation pressure should equal
crC = F ~ +~mv
Likewise the pressure at the time of failure can be
determined in other ways than using the failure load
container 50 and the failure load 52 of Figure 3. Alter-
native devices for applying the failure pressure can be
employed consistent with the test method, and so long as
the measured pressure at f~ilure does not exceed the
following quantity, then flow will occur by gravity
alone:
.
~ CrF = ( D--I) C ~ "
. . ,
` 25 A second preferred embodiment is shown in Figure S
and 6, during the consolidation phase and at the begin- !'
ning of the failure phase of the test procedure respec-
tive~ly. In this embodiment, the other parts of the '
,~ .
~` i
. . i

-15- ~2~ 3~
apparatus are supported above the floor 88 or bench
top by a bottom spacer ring 100.
The test cell in this embodiment includes a ring ~2
that has a conical inner surface 94. During the con-
solidation phase, the ring 92 is supported on the plugsupport ring 96, which in turn rests on the bottom spacer
ring 100. As shown in the drawings, the height of the
ring 92 is C, and D represents the diameter of the larger
end of the conical surface 94.
The plug 90, which in one version of this embodi-
ment includes the cylindrical spacer 102, spans the lar~er
diameter end of the conical surface 94 and rests on a
beveled portion of the plug support ring 96. In addition
to supporting the plug 90, the beveled portion of the
lS ring 96 centers the plug 90. The concave conical
surface 98 of the plug 90 faces the interior o~ the test
chamber. In accordance with the present invention, the
inwardly-facing surface of the plug 90 must be concave,
but it does not necessarily have to be conical.
A mold ring 104 rests on the ring 92. The inside
surface of the mold ring is cylindrical and its diameter
is the same as the diameter of the smaller end of the
conical surface 94, so that the mold ring 104 forms an
upward extension of the ring 92.
The consolidation pressure is applied, in one
version of this embodiment, by applying a weight to the
consolidation disk 106 that rests on the particulate
- material 112. The disk 106 has a diameter slightly smaller
than the inside diameter of the mold ring 104, to permit
the disk 106 to move freely in the axial direction as the
material 112 compacts. In this version of the embodi-
ment, a spacer 114 supports a consolidation load con-
tainer 108 a short distance above the disX 106. The

-16-
consolidation load may be increased by adding con-
solidation ballast 110 to the consolidation load
container 108.
During the consolidation phase, the direction of
movement of the particles is downward toward the plug 90.
The downwardly-diverging shape of the conical surface 94
prevents friction between that surface and the particulate
material from interfering with the movement of the
particles.
The consolidation load should be applied to the
particulate material in the test cell for an interval of
time approximately equal to the time at rest of the
particulate material in the full-size hopper. If the
material is moist and if the time is long, a moisture
retaining membrane similar to the membrane 40 of
Figure 2 may be employed to prevent loss of moisture.
A~ the end of the consolidation phase, the load is
removed. Next the mold ring 104 is removed and the con-
solidated material above the ring 92 is removed.
Thereafter, the rings 92 and 96 along with the
plug 90 are inverted, care being taken to prevent
relative movement of the rings 92 and 96, and to avoid
bumping the exposed portions of the plug 90. The ring 92
is gently lowered onto the bottom spacer ring 100, so that
the apparatus then has the configuration shown in
Figure 6. The failure load container 116, which may be
identical to the consolidation load container 108, is
then placed on the spacer 102. The load on the parti-
culate material is gradually increased by adding ballast
to the failure load container 116, until failure occurs.
The load at which failure occurs is then noted.
Only a small downward movement of the plug 90
occurs before failure, and the failure is catastrophic,

with the particulate material abruptly falling from the
ring 92 onto the floor 88 or bench top. To accom-
modate this small motion of the plug 90, the outside
diameter of the plug 90 is slightly less than the diameter
S of the larger-diameter end of the surface 94. This is
shown somewhat exaggerated in ~igures 5 and 6 for
clarity. Likewise, the wall thickness of the plug 90 is
exaggerated in the drawings.
During the failure phase of the test procedure,
the downwardly converging surface 94 simulates the
downwardly converging walls of the full-size hopper.
As was the case with the first preferred embodiment
described above, by the use of appropriate pre-calculated
consolidation loads, any of several flow-related
properties of the particulate material may be ascertained
by use of the apparatus of this second preferred embodi-
ment. The equations for the apparatus of this second
preferred embodiment are identical to those used in the
first preerred embodiment described above.
Thus, there has been described a novel apparatus
for scale-model testing to determine whether or not flo~l
will occur under the action or gravity from a circular
or a rectangular outlet in the bottom of a container
that is partly filled with particulate matter. The
shape of the space within the test cell is of particular
significance. The test chamber walls conform to the
~ur~ace of a cone, and the end of the test chamber is
plugged by a movable plug having a concave inwardly-
facing surface. Accordingly, the material under test
never contacts a cylindrical wall and this redu~es test
error caused by the unknown frictional forces that come
into play when the material contacts a cylindrical wall.
Also, the shape of the test cell is designed to not

z
-18-
interfere with the plastic stress field that develops
in the sample during testing. Accordingly, the novel
shape of the test cell is highly advantageous and results
in greater accuracy.
There has also been described in detail a scale
model test procedure that allows accurate prediction of
whether or not an outlet in the bottom of a ~ontain~r oF
particulate matter will discharge under the action of
gravity alone. This method, which requires the use o~
the special test apparatus involves the steps of con-
solidating the material to a specific extent by appli-
cation of a specific pressure to it and then inverting
the test cell prior to determining the pressure required
to cause failure of the material.
Several embodiments and variations have been des-
cri~ed in detail above by way of illustration. Further
variations will, no doubt, be apparent to those skilled
in the art, and such further variations are regarded as
being within the scope of the invention.
Industrial Applicability
The method and apparatus described above is most
helpful in predicting the behavior of full-scale con-
tainers of particulate materials from small-scale
laboratory bench tests. In this way the success of a
hopper design can be assured and expensive mistakes can
be avoided. The method is applicable to all kinds of
particulate materials, from the finest powders to coarse
ores, so that the potential applications are innumerable.

Representative Drawing