EFFECT OF HOPPER ANGLE ON FLOW OF GRANULAR
MATERIALS THROUGH RECTANGULAR ORIFICES
by
CHIEN KUEI TSAI, B.S.
A THESIS
IN
CHEMICAL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
IN
CHEMICAL ENGINEERING
Approved
Accepted
December, 1991
ACKNOWLEDGEMENTS
~ am deeply indebted to Dr. Raghu s. Narayan, chairman of
my advisory committee, for his advice and guidance during the
write up of my thesis and his suggestion on the presentation
of my research. He also gave me full freedom in my research.
Sincere appreciation extended to Dr. Clifford B. Fedler,
for his patiently and timely assistance in the accomplishment
of my work; to Dr. Jim M. Gregory, for his guidance and
enthusiasm in the development of the research model; to Dr.
Richard w. Tock, for his sincere suggestion and criticism.
I owe special thanks to Jim Snyder and Byron Neal who
patiently worked with me building the research equipment in
the summer of 1990; to Tammy Kent and Dawn Eastman, for their . assistance in preparing of my thesis.
To my Lord and Savior, Jesus, for changing my life,
assuring me the purpose of my presence in this world, and
working with me through the happy and difficult times of this
work. Most of all, I thank my family, back home in Taiwan,
and many friends for their constant encouragement and support.
ii
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
LIST OF TABLES iv
LIST OF FIGURES vi
CHAPTER
I. INTRODUCTION 1
II. LITERATURE REVIEW 4 2.1. Historical Overview 4 2.2. The Effect of Hopper Angle
on Granular Flow 19 2.3. Correlations of Gregory and Fedler 25
III. EXPERIMENTAL SETUP AND PROCEDURE 28 3.1. The Model Hopper 28 3.2. The Procedure 31 3.3. The Granular Materials 34
IV. MODEL DEVELOPMENT 3 6
V. RESULTS AND DISCUSSION 45 5.1. Visual Observation of Flow Phenomena 45 5.2. Bulk Solids 47 5.3. Powders 82
VI. CONCLUSIONS AND RECOMMENDATIONS 111
BIBLIOGRAPHY 113
APPENDIX 124
iii
LIST OF TABLES
3.1. Values of material angle properties 33
3.2. Values of material bulk density and flow resistance factor 35
4.1. Coefficient A1 of each material used in equations (4.5) and (4.6) 40
4.2. Coefficient A2 of each material used in equations (4.5) and (4.6) 41
5.1. Values of adjustment factor for polypropylene at various hopper angles 52
5.2. Values of adjustment factor for low density polyethylene at various hopper angles 59
5.3. Values of adjustment factor for reground nylon 6 at various hopper angles 65
5.4. Values of adjustment factor for dry sorghum at various hopper angles 72
5.5. Values of adjustment factor for cornmeal at various hopper angles 89
5.6. Values of adjustment factor for linear low density polyethylene at various hopper angles 95
5.7. Values of adjustment factor for polyphenylene sulfide at various hopper angles 102
A.1. Mass flow rate of polypropylene at various orifice sizes and hopper angles 125
A.2. Mass flow rate of low density polyethylene at various orifice sizes and hopper angles 126
A.3. Mass flow rate of reground nylon 6 at various orifice sizes and hopper angles 127
A.4. Mass flow rate of dry sorghum at various orifice sizes and hopper angles 128
iv
A.5. Mass flow rate of polypropylene at various orifice sizes and hopper angles
A.6. Mass flow rate of cornmeal at various orifice sizes and hopper angles
A.7. Mass flow rate of linear low density polyethylene at various orifice sizes and hopper angles
A.B. Mass flow rate of polyphenylene sulfide at various orifice sizes and hopper angles
v
129
130
131
132
LIST OF FIGURES
2.1. Properties cited in literature which affect flow of granular materials
3.1. Front and top views of the model hopper
4.1. Plot of A, coefficient of LOPE, PP, dry sorghum, LLDPE, and PPS as a function of orifice opening
4.2. Plot of A1/K of LOPE, PP, LLOPE, PPS, and dry sorghum as a function of orifice opening
5.1. Plot of mass flow rates of PP as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.2. Plot of average adjustment factors of PP as a function of hopper angle
5.3. Plot of adjustment factors of PP as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.4. Plot of predicted versus measured mass flow rate of PP
5.5. Plot of mass flow rates of LOPE as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.6. Plot of average adjustment factors of LOPE as a function of hopper angle
5.7. Plot of adjustment factors of LOPE as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.8. Plot of predicted versus measured mass flow rate of LOPE
5.9. Plot of mass flow rates of reground Nylon 6 as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.10. Plot of average adjustment factors of reground Nylon 6 as a function of hopper angle
vi
18
29
42
43
48
50
51
53
55
56
57
60
61
63
5.11. Plot of adjustment factors of reground Nylon 6 as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.12. Plot of predicted versus aeasured mass flow rate of reground Nylon 6
5.13. Plot of mass flow rates of dry sorghum as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.14. Plot of average adjustment factors of dry sorghum as a function of hopper angle
5.15. Plot of adjustment factors of dry sorghum as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.16. Plot of predicted versus measured mass flow rate of dry sorghum
5.17. Plot of A1 coefficient of PP as a function of orifice opening size
5.18. Plot of A1 coefficient of LOPE as a function of orifice opening size
5.19. Plot of A1 coefficient of reground Nylon 6 as a function of orifice opening size
5.20. Plot of A1 coefficient of dry sorghum as a function of orifice opening size
5.21. Plot of ~ coefficient of PP as a function of orifice opening size
5.22. Plot of~ coefficient of LOPE as a function of orifice opening size
5.23. Plot of ~ coefficient of reground Nylon 6 as a function of orifice opening size
5.24. Plot of ~ coefficient of dry sorghum as a function of orifice opening size
5.25. Plot of mass flow rates of flour as a function of hopper angle at orifice opening of 8.9 to 15.2 em
vii
64
66
68
70
71
73
74
75
76
77
79
80
81
82
84
5.26. Plot of mass flow rates of cornmeal as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.27. Plot of average adjustment factors of cornmeal as a function of hopper angle
5.28. Plot of adjustment factors of cornmeal as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.29. Plot of predicted versus measured mass flow rate of cornmeal
5.30. Plot of mass flow rates of LLDPE as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.31. Plot of average adjustment factors of LLDPE as a function of hopper angle
5.32. Plot of adjustment factors of LLDPE as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.33. Plot of predicted versus measured mass flow rate of LLDPE
5.34. Plot of mass flow rates of PPS as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.35. Plot of average adjustment factors of PPS as a function of hopper angle
5.36. Plot of adjustment factors of PPS as a function of hopper angle at orifice opening of 2.5 to 15.2 em
5.37. Plot of predicted versus measured mass flow rate of PPS
5.38. Plot of A1 coefficient of cornmeal as a function of orifice opening size
5.39. Plot of A1 coefficient of LLDPE as a function of orifice opening size
5.40. Plot of A1 coefficient of PPS as a function of orifice opening size
viii
85
87
88
90
91
93
94
96
98
99
100
103
104
105
106
5.41. Plot of ~ coefficient of cornmeal as a function of orifice opening size
5.42. Plot of ~ coefficient of LLDPE as a function of orifice opening size
5.43. Plot of ~ coefficient of PPS as a function of orifice opening size
ix
107
108
109
CHAPTER I
INTRODUCTION
The flow of granular materials (i.e., bulk solids and
powders) through orifices in a storage vessel such as hoppers
and bins, with inclined walls is a very common and important
industrial unit operation. Most industries get involved at
one point or another with processing, handling, or storing
granular materials. These industries include chemical polymer
production, pharmaceutical tablet and powder manufacturing,
mineral mining, agricultural grain and food processing, and
glass and plastics manufacturing. Millions of tons are
involved annually and yet little has been done to study these
granular materials in relation to gravity flow through
orifices.
Primary material transport for flow control operations in
such industries is accomplished by flow through orifices.
Usually materials are withdrawn from a storage bin by allowing
the materials to flow under the action of gravity through an
outlet in the bottom of the bin. Unfortunately, many of these
bulk solids and powders do not flow reliably or uniformly
through hoppers, storage bunkers, bins, chutes, surge vessels,
feeders, stockpiles, or reactors. Although in the worst cases
complete stoppage of flow may occur due to stable bridging or
channelling, in many situations the flow is maintained but is
rather unstable. The results are loss in production, extra
1
labor, plant downtime, poor quality control,
materials, and unreliable processing.
spoiled
As a result, information on the flow rate of granular
materials through orifices is needed to determine flow and
properly size the opening for flow control during transfer of
materials. In order to properly design systems involving
granular flow, a number of researchers and engineers have
attempted to develop mathematical models that could adequately
describe the flow process. Despite of these efforts, the
design and control of granular material transfer continue to
be more of an art than a science.
The behavior of granules in gravity flow through hoppers
under the influence of wall inclination (or hopper side wall
angle) is also an area of research which has received
surprisingly little attention in the past, inspite of the
significant application potential in process industries. Due
to the inadequacy of the existing theories and the lack of
experimental data, the problem involving granular flow has
never been satisfactorily solved. However, industries still
proceed largely by trial and error in the design of hoppers or
bins in a material handling system, an approach that is
inconsistent with the level of technology that has· been
developed in other areas.
2
The objectives of this research were:
(1) Develop a uniform mathematical model to predict the flow
of granular materials, using an existing physically based
prediction equation as a starting framework,
(2) Specify the limitations based on the scope of
experimental work,
(3) Evaluate and. model the effect of wall inclination (or
hopper angle),
(4) Evaluate and model the effect of orifice size on the flow
rate of both cohesive and non-cohesive granular materials
through horizontal rectangular orifices in a hopper
device.
3
CHAPTER II
LITERATURE REVIEW
2.1 Historical Overview
Flow of granular materials has existed since the early
time of mankind. The earliest example of granular flow is the
sand in an hourglass. According to Nedderman et al. (1982),
model developments began in the mid 1800's when the flow rate
for a circular orifice was described as a function of the
diameter of the opening. Hagen ( 1852) was the first to
initiate a formal investigation of granular flow phenomena.
He used dimensional analysis to correlate data on the flow
rate of sand through circular orifices of different sizes.
Development of a prediction equation to describe the flow
of granular materials through orifices was attempted more than
seven decades ago. However, much of the early work on flow in
bins was associated with research on wall pressure only.
The earliest comprehensive work on storage of granular
materials was done by Ketchum (1919). He summarized the work
of several investigators during the 1890 to 1915 period, as
well as reporting on his own experiments. The major
objectives for these early studies, cited in Ketchum's.book,
was the need to store large quantities of grain. He used
wheat and showed that the flow was independent of head and
suggested that flow varied as the cube of the orifice
4
diameter. However, nothing was mentioned about the effect of
hopper angle on the flow rate.
OVer the next 30 years a few papers seemed to have been
published on grain pressure and grain flow based on Janssen's
theory (1895) of stress analysis in which he recognized the
major difference between a fluid and a powder. As a powder
can support shear stress, part of the weight of a stored
material is transferred to the walls. A number of people had
followed his theory or developed their own theory on pressure
or stress analysis (Shaxby and Evans, 1923: Saul, 1953:
Shinohara and Tanaka, 1974: Spink and Nedderman, 1978:
Nakajima et al., 1985). However, Janssen's equations, using
constant values of friction, was shown to be unable to predict
the loads for changing filling heights (Calil and Haaker,
1989). It was also well-known that bunker design according to
the simple Janssen's theory may sometimes fail unexpectedly
(Blair-Fish and Bransby, 1973).
Beginning in the mid 1950's, there was a great deal of
interest in flow patterns and flow rates from both flat bottom
bins and conical bins (Rose and Tanaka, 1959: Athey et al.,
1966: Gardner, 1966: Novosad and Surapati, 1968: Blair-Fish
and Bransby, 1973: Lee et al., 1974: VanZanten et al., '1977:
Nedderman et al. , 1982: Tiiziin and Nedderman, 1982:
Michalowski, 1984: Michalowski, 1987: Standish and Liu, 1988:
Peterson, 1989: Bucklin et al., 1991). Flow patterns were
found to be very sensitive to the degree of compaction of the
5
material. However, no theoretical method was developed that
adequately predicted the observed flow pattern. Moreover,
based on the study of plastic powders, Farley and Valentin
(1965) stated that the flow pattern for homogeneous materials
(i.e., one type of material with similar particle size and
shape) was not so important and the only requirement was a
constant rate of withdrawal and a certain live capacity (i.e.,
area of the storage vessel where flow takes place).
Jenike (1961, 1964) developed the theory of quasi-static
analysis for determining the proper design dimensions for mass
flow from bunkers and silos. The basics for his theory is
that gravity flow of a solid in a bin will occur, provided the
yield strength of the solid is insufficient to support an
obstruction to flow. The material yield strength is a
function of the compacting pressure which is in turn a
function of the material position within the bin. Bin design
procedures were based on predicting the yield strength of the
material using a set of equilibrium equations as the solids
flowed through the bin. Jenike solved these differential
equations for a hopper with a radial stress field in which the
mean stress in the material was assumed to vary linearly with
the radial distance from the center of the hopper. However,
no unique velocity field could be derived from such a quasi
static analysis.
Since then, many researchers have either used or extended
Jenike's theory to solve problems associated with granular
6
flow (Johanson and Colijn, 1964; Walker, 1966: Bruff and
Jenike, 1967; Walters, 1973; Enstad, 1975; Richards, 1977; Van
Zanten and Mooij, 1977; Van Zanten et al., 1977; Johanson,
1978; Nguyen et al., 1979; Johanson and Cox, 1989). However,
most of these studies were concerned with getting materials to
flow, not with rate of flow. Moreover, Kaza and Jackson
(1984) demonstrated that the picture of motion of Jenike's
theory was inconsistent with the law of motion. Studies by
other researchers (Walker and Blanchard, 1967; Keno and Huang,
1989) have also questioned the validity of Jenike's velocity
profile or flowability as the general design criteria.
Besides Jenike's bunker and silo design, Lee (1960) ran
experiments on the flow of coal and developed a large scale
hyperbolic hopper design. However, Lee's theory did not
satisfactorily account for the effect of material cohesion.
Other theoretical hopper designs have also been developed
(Gardner, 1963; Sigley and Chaplin, 1982).
Granular flow has also been observed to be different from
liquid flow since the flow rate was found to vary
theoretically with the orifice diameter to the power of 2. 5 to
3.0 (Gregory and Fedler, 1987). Most studies also had their
values in this range. Other researchers (Chang et al.; 1984
and Gregory and Fedler, 1987), however, had identified
granular flow to have some characteristics similar to fluid
flow. Newton et al. (1945) even developed an equation which
7
included a head (or material depth) term to describe qranular
flow throuqh orifices. Their equation was qiven as:
(2 .1)
where MP is the predicted mass flow rate, D is the orifice
diameter, and H is the head of packinq above the orifice.
However, this relationship to head is very small comparinq to
other parameters, such as orifice diameter. Many researchers
have stated that flow rate of qranular materials throuqh an
openinq is independent of the depth above the openinq (Fowler
and Glastonbury, 1959; Ewalt and Buelow, 1963; Sullivan, 1972;
Davidson and Nedderman, 1973; Nquyen et al., 1979; Kaza,
1982; Chanq et al. , 1984) • In qeneral, most researchers aqree
that head does not affect flow until it is equal to or less
than the orifice diameter. While some characteristics are
similar, it is qenerally accepted that qranular flow does not
exactly follow the laws of hydrodynamics. This has confirmed
the observation of Ketchum (1919) reqardinq the neqliqible
effect of head on flow.
Several models had been developed based on semi-empirical
methods or dimensional analysis (Deminq and Mehrinq, '1929;
stahl, 1950; Franklin and Johanson, 1955; Sinqley, 1958;
Fowler and Glastonbury, 1959; Beverloo et al., 1961; Ewalt and
Buelow, 1963; Kotchanova, 1970; Chanq et al., 1984; Moysey et
al., 1988). The primary focus of these models has been the
8
prediction of flow through horizontal orifices. Survey of
physical principles of granular flow as well as careful
observation of their own experiments have been done by several
researchers (Hermans, 1953; Deresiewicz, 1958; Stepanoff,
1969; Brown and Richards, 1970; TUzun et al., 1982;
Shahinpoor, 1983; stanley-Wood, 1983; Fayed and Otten, 1984;
Cheremisinoff, 1986).
Based on their studies in 1959 and 1960, Brown and
Richards (1970) drew considerable attention to the concept of
"empty annulus," i.e. , a concentric ring where there is
essentially no flow. According to their observation, not all
of the orifice area is utilized during flow because particles
adjacent to the opening overhang it slightly and thus reduce
its effective size. In other words, no particle can approach
within a distance of one-half the particle diameter (i.e. , the
width of the empty annulus) to the orifice edge. Because of
the inward motion of particles toward the central axis,
contraction of the flow continues below the exit opening to a
minimum cross-section, called vena contracta. The same
phenomena is observed in fluid flow.
Franklin and Johanson (1955} obtained data for the flow
of various granular materials, such as glass beads,· lead
shots, and sand, through circular orifices. Their correlation
took into account the concept of empty annulus and the angle
of friction. They were able to correlate their results using
the following expression:
9
p ·D2•93
COSCZ
(6.288·tancz + 23.16) ·(d.P + 1.889) - 44.9 '1 + coscz <2•
2>
where Pp is the particle density of the material, ~ is the
equivalent particle diameter, and • is the emptying angle of
repose. The disadvantage of this correlation was that it is
dimensionally inconsistent and cannot predict the data of
other orifice shapes very well.
Fowler and Glastonbury (1959) studied the flow of wheat,
sugar, rice, rape seed and sand through both circular and non-
circular horizontal orifices and found that the flow rate
could be predicted from the following correlation:
D M = 0. 236 •p ·A·.J2 ·g-D · ( __lL) o.us ·-p b IJ "l'd..P
(2.3)
where pb is the material bulk density, A is the effective
orifice area, ~ is the hydraulic diameter, ~ is the
sphericity, and g is the gravitational constant.
Beverloo et al. (1961) took the concept of empty annulus
into account by reducing the orifice diameter in .their
correlation and became the first most widely accepted equation
for the prediction of the discharge rate of granular materials
from an orifice. They studied the flow of a number of
materials, primarily seeds, through various orifices and
10
reported that the flow rate could be estimated from the
following equation:
(2.4)
where n8 is a coefficient with a value between 1.3 and 1.5.
In this equation, actual orifice diameter utilized by the flow
is (0- n8·~). However, this equation does not explain why
ns is larger than 1. Thus I the theory that describes the flow
was not very well defined.
Ewalt and Buelow (1963) studied the flow of dry shelled
corn through rectangular and circular orifices in bins without
considering the effect of empty annulus. Their mathmetical
correlations are the most basic of those studied and are given
as:
Q = n:11111 ·D12.a for circular orifices (2.5)
Q = n83·L,...·wD- for rectangular orifice (2. 6)
where Q is the volumetric flow rate, L is the length of the
slot, w is the width of the slot, and coefficients nE81 , nEB2 ,
nEB3 , nEB4 , nEBS are calculated to be 0.1196, 3.01, 0.1531,
1. 62 and 1. 4. However, these coefficients will vary for
different materials. Their correlations were limited to
orifice openings of 13 em or less and for dry shelled corn
only.
11
Though many researchers have explored the subject of
granular flow under different assumptions, Fedler (1988) has
reviewed and evaluated several models and found the following:
the Fowler and Glastonburg (1959) model (equation 2.3)
contained a constant that has to vary for different materials
in order to obtain a reasonable fit to measured data: the
Beverloo (1961) model (equation 2.4) contained a coefficient
that is not a constant and has to be determined for various
materials: and the Ewalt and Buelow (1963) model (equations
2.5 and 2.6) developed by using a power equation was simple,
but the coefficients had no physical significance. The study
of ·Sarkar et al. (1991) have also shown that granular flow
rate depends on a number of variables in addition to those
considered by Beverloo et al. (1961). Therefore, it is not
possible to use these models without calibration.
Various researchers (Craik and Miller, 1958: Corn, 1961;
Chang et al., 1984; Moysey et al., 1988; Gregory and Fedler,
1987; Fedler and Gregory, 1989; Kaye, 1989) have shown that
flow rate varies with material properties. Chang et al.
(1984) studied the flow of corn through orifices and found
that the effect of moisture on the change in flow rate was
negligible when the moisture content is less than 19% (wet
basis). Results of many studies (Brusewitz, 1975: Fickie et
al., 1989: Hyun and Spalding, 1990) also indicated that,
instead of flow rate, bulk density was the main parameter
affected by moisture. In addition, studies of Fedler (1988)
12
concluded that any model that excluded the bulk density term
could not predict flow to a high degree of accuracy without
extensive laboratory testing and statistical curve fitting of
the data.
Moysey et al. (1985) studied the flow of wheat, flax,
barley and rapeseed through orifices and suggested that
particle size, sphericity and surface roughness all contribute
to differences in flow rate. Though no fundamental cause and
effect relationship was given to explain the effect of these
variables on flow rate, general material property parameters
such as angle of repose, density, shape (or sphericity), and
particle size, were used to correlate granular flow rate
through orifices. Angle of repose is defined as the angle to
the horizontal assumed by the free surface of a heap at rest
and obtained under pouring (i.e., filling) or draining (i.e.,
emptying) conditions. However, Fedler and Gregory (1989)
found that both angle of repose and sphericity have no
significant correlation with mass flow of dry granular
materials through horizontal orifices. Rather, particle size
and surface roughness (or irregularity of the granule) were
the primary factors affecting flow resistance, and thus the
flow rate.
Several researchers (Franklin and Johanson, 1955: Fowler
and Glastonburg, 1959: Ewalt and Buelow, 1963: Chang et al.,
1984: Chang and Converse, 1988) have developed flow prediction
models based on statistical curve fitting techniques with good
13
results. However, recal ibration of the equations is necessary
for new materials. Other methods based on a perturbation and
coordination expansion with respect to the hopper angle to
allow for the effect of gravity and rough walls were used for
an approximate solution (Savage, 1967: Brennen and Pearce,
1978: Nguyen et al., 1979). A number of workers have
approached granular flow problems by attempting to solve a set
of ordinary or partial differential equations (Delaplaine,
1956: Davidson and Nedderman, 1973: Williams, 1977: Brennen
and Pearce, 1978; Savage and Sayed, 1979: Nguyen et al., 1979:
Kaza and Jackson, 1982: Chen et al., 1984: Schaeffer, 1987:
Prakash and Rao, 1988: Mountziaris, 1989), though very little
is known about the nature of the solutions to these equations.
A number of researchers have developed kinetic models for
flow of coarse granular materials in flat bottom hoppers
(Ahmadi and Shahinpoor, 1983: Haff, 1983: Jenkins and Savage,
1983: Jackson, 1986: Graham et al., 1987: Lun and Savage,
1987: Ma and Ahmadi, 1988; Abu-Zaid and Ahmadi, 1990). The
kinetic models result in equations which can be easily solved
and successfully predict the velocity profile only in the
converging zone just above the orifice. On the other hand, '
many researchers used classical plasticity theory to predict
the velocity profile of granular flow (Nemat-Nasser, 1983:
Polderman et al., 1987: Han et al., 1989). It is believed
that only part of the material in the hopper experiences
plastic deformation and, therefore, traditional plasticity
14
theory does not seem to be a promising frame of analysis.
Others have developed scale models to study granular flow of
coal in chutes and silos (Wolf and Hohenleiten, 1948: Carson
and Royal, 1989), though no reliable scaling criteria have
been demonstrated. Soo et al. (1989) used recirculating pipe
flow of dense suspension to measure the average velocity of
glass beads.
A number of studies on granular flow have been concerned
more with the mechanical behavior of the particles, rather
than to predict the flow rate. Chatlynne and Resnick (1973)
used a freezing technique to photographically study flow of
colored glucose spheres through flat-bottomed bins. Kono et
al. (1989) studied the effect of flow conditioners, such as
polysaccharides and calcium stearate powders, on the
flowability of cohesive coal powder. Standish and Liu (1988)
applied a residence time distribution (RTD) theory on granular
flow problems. Molerus (1978) studied the effect of inter
particle cohesive forces on the flow behavior of barytes
powders. Kott and Kramer (1966) studied the influence of wall
material or surface coating on the flow of powdered clay
through bins. Iverson and LaHusen (1989) studied the
intergranular pore pressure fluctuation of fiberglass rods in
a bin filled with water. Others (Jones and Pilpel, 1966:
Farley and Valentin, 1968: Danish and Parrot, 1971: Carstensen
and Chan, 1977: Nikolakakis and Pilpel, 1988: Arteaga and
Tiiziin, 1990) have studied the effect of material or size
15
mixture on the flow of granular materials and found that
undesirable segregation and irregular flow pattern occurred
during the flow process.
A number of people have used computer animation to study
the velocity and pressure profiles of granular flow. However,
at present, computer simulation is too time-consuming,
cumbersome and incomplete to describe such interactions in
three dimensions.
Besides extensive studies on granular flow through
symmetric geometries, various methods have been used to
investigate flow behavior in asymmetric geometries (Takahasi,
1937: Gardner, 1964: Novosad and Surapati, 1968: Giunta, 1966:
Van Zanten and Mooij, 1977: Van Zanten et al. , 1977) • Most of
these methods suffered from being cumbersome and expensive and
were only suitable for use on a small scale. In addition,
some researchers have extended their results to vertical
orifices (Ewalt and Buelow, 1963: Fedler and Gregory, 1988:
Chang et al., 1990) or inclined pipes, chutes, or walls
(Takahasi, 1937: Gardner, 1962: Drake, 1986: Sarkar et al.,
1991).
Many researchers have applied the theories of granular
flow to various chemical process operations. For example,
Benkrid and Caram (1989) applied the kinetic theory of
granular flow in a spouted bed: likewise, Ding and Gidaspow
(1990) applied the kinetic theory to a bubbling fluidization
16
model. Lamptey and Thorpe (1991) studied the discharge of
solid-liquid mixture from hoppers using the hourglass theory
of Davidson and Nedderman (1973) for the solid flow, coupled
with the Bernoulli theorem and Ergun equation for the liquid
flow. Many researchers (Rausch, 1949; Leung and Jones, 1978;
Ginestra et al., 1980; Chen et al., 1984; Mountziaris and
Jackson, 1991) have also studied granular flow through
orifices in vertical standpipes which are often fed by a feed
device, normally a hopper with walls steep enough to permit
its entire content to be in motion when discharging (called
funnel flow).
Based on the information available in the literature,
there were general agreements and disputes among most workers
about the fundamental principles governing flow behavior as
well as the significance of some of the parameters used to
describe the flow. Though the subject of granular flow has
received considerable theoretical and experimental attention
for some time, there is still no model that can be use to
describe all areas of the flow problems and there does not
exist a universal correlation that can predict the flow
accurately for all ranges of geometries and granular
materials. Properties cited in the literature, which 'might
affect the flow of granular materials, are listed in Figure
2.1.
17
Single Particle Parameters Bulk Particle Parameters
Density Adhesion
Hardness Bulk Density
Hygroscopicity Cohesion
Light Scattering/Adsorption Conductivity
Shape Electrostatic Charge
Size Moisture Content
Surface Pore Size Distribution
Porosity
Shear Strength
Size Distribution
Surface Area
Compressive Strength
Figure 2.1. Properties cited in literature which affect flow of granular materials
18
~.2 The Effect of Hopper Angle on Granular Flow
The effect of hopper angle on flow of granular materials
through orifices has been known from very early times.
However, many researchers have proposed methods for predicting
granular flow rates without taking into account the effect of
wall friction, which in turn was affected by the hopper angle.
As a result, the solutions obtained by such methods did not
correctly predict the effect of the hopper angle on the
discharge flow rate. In addition, most resec~rchers have
mentioned in one way or another that the hopper angle should
be large enough to permit desirable flow, but without a
quantitative correlation describing its effect on the flow
rate in their granular flow models (Jenike, 1964: Johanson and
Coijn, 1964: Farley and Valentin, 1965: Chen et al., 1984).
For example, Wolf and Hohenleiten (1948) studied the flow
of coal through a hopper model having different values of
orifice openings and side wall slope to the horizontal.
Results of their studies showed that hoppers with steeper side
walls can handle coal with high cohesion and moisture content,
and a designed side wall slope equal to or greater than 70° is
generally acceptable. However, no function of the side wall
slope on flow rate was presented.
Nguyen et al. (1980) identified a number of specific flow
types (i.e., mass flow and several forms of funnel flow) from
experimental observation of granular flow in hoppers.
Transitions between these types of flow were found to depend
19
mainly on the hopper angle and other properties such as width
of the exit opening and frictional properties of the granular
materials and the side walls. Their results show the
importance of the hopper angle on flow, though no correlation
was given.
Probably the earliest study on granular flow including
correlations to describe the effect of hopper angle on flow
rate was done by Deming and Mehring (1929) who used the angle
of sliding as their primary parameter. The angle of sliding
is defined as the angle to the horizontal of an inclined
surface on which an amount of granular material will slide
downward due to the influence of gravity and surface friction.
Since then, various workers have studied the effect of wall
inclination on granular flow rate.
Takahasi (1937) investigated the flow rate of sand with
respect to the inclination of a wooden canal. Based on his
findings, he concluded that there were two kinds of modes in
the sand flow: laminar and turbulent flows. The laminar flow
occurs when the upper layer of the material on the side walls
slides like a solid plate, and the lower layer of the
materials fixes on the surface of the side walls and rotates
itself. The turbulent flow occurs when each particle runs on
its own independently and violently in different directions
downwards. These two modes of flow were determined by the
wall inclination (i.e., hopper angle). The mechanism of each
kind has also been studied using high speed photographic
20
techniques. These two kinds of flow, laminar versus
turbulent, were later confirmed by Gregory and Fedler (1987).
A major contribution to the understanding of the effect
of hopper angle on the flow rate was the work of Rose and
Tanaka (1959). They studied several materials flowing in a
small-scale cylindrical hopper with conical bases for the
effect of cone angle (i.e., hopper angle to the horizontal) on
the flow rate. For a small value of cone angle, a stagnant
region will occur and it is reasonable to assume that since
the side walls will be buried within the stagnant material,
hopper angle will have little effect on the flow. Following
this idea, they proposed that flow rate is a function of
(tan9tan~)-0 • 35 when the cone angle is greater than the angle
of repose. Their equation was given as:
n ·pb·D2 • 5 ·y'g· (0. 505 - 0.16 tant)
~ = 4·(tan9·tan~) 0 • 35 (2.7)
where e is the hopper angle to the horizontal, and • is the
angle of internal friction. Their results had shown that
effects on the discharge rate by the variation of the cone
angle forming the bottom of the bin would be different
according to whether the cone angle was greater or less than
the angle of repose.
Gardner (1962) has developed an expression to calculate
the limiting thickness of granular materials which will flow
down an inclined wooden plane based on the limiting span
21
mainly on the hopper angle and other properties such as width
of the exit opening and frictional properties of the granular
materials and the side walls. Their results show the
importance of the hopper angle on flow, though no correlation
was given.
Probably the earliest study on granular flow including
correlations to describe the effect of hopper angle on flow
rate was done by Deming and Mehring (1929) who used the angle
of sliding as their primary parameter. The angle of sliding
is defined as the angle to the horizontal of an inclined
surface on which an amount of granular material will slide
downward due to the influence of gravity and surface friction.
Since then, various workers have studied the effect of wall
inclination on granular flow rate.
Takahasi (1937) investigated the flow rate of sand with
respect to the inclination of a wooden canal. Based on his
findings, he concluded that there were two kinds of modes in
the sand flow: laminar and turbulent flows. The laminar flow
occurs when the upper layer of the material on the side walls
slides like a solid plate, and the lower layer of the
materials fixes on the surface of the side walls and rotates
itself. The turbulent flow occurs when each particle rUns on
its own independently and violently in different directions
downwards. These two modes of flow were determined by the
wall inclination (i.e., hopper angle). The mechanism of each
kind has also been studied using high speed photographic
20
techniques. These two kinds of flow, laminar versus
turbulent, were later confirmed by Gregory and Fedler (1987).
A major contribution to the understanding of the effect
of hopper angle on the flow rate was the work of Rose and
Tanaka (1959). They studied several materials flowing in a
small-scale cylindrical hopper with conical bases for the
effect of cone angle (i.e., hopper angle to the horizontal) on
the flow rate. For a small value of cone angle, a stagnant
region will occur and it is reasonable to assume that since
the side walls will be buried within the stagnant material,
hopper angle will have little effect on the flow. Following
this idea, they proposed that flow rate is a function of
(tan9tan·)-0 · 35 when the cone angle is greater than the angle
of repose. Their equation was given as:
n ·pb·D2 • 5 ·y'g· (0. 505 - o .16 tant) ~ = 4·(tan9·tan~) 0 • 35
(2.7)
where e is the hopper angle to the horizontal, and • is the
angle of internal friction. Their results had shown that
effects on the discharge rate by the variation of the cone
angle forming the bottom of the bin would be different
according to whether the cone angle was greater or less than
the angle of repose.
Gardner (1962) has developed an expression to calculate
the limiting thickness of granular materials which will flow
down an inclined wooden plane based on the 1 imi ting span
21
theory of Richmond and Gardner ( 1962) . He concluded that flow
would not occur in either case if the angle of inclination
(i.e., hopper angle to the horizontal) was less than both the
angle of internal friction of the materials and the angle of
sliding of material on the walls. Thus he proposed that the
minimum angle of inclination for which flow can occur is
controlled by the larger of (\·~- ¢) and (~·~- ~) where pis
the angle of sliding to the horizontal.
Ewalt and Buelow (1963) conducted experiments in bins
with horizontal orifice openings by taking into account both
the angles of internal and wall friction. Their results
indicated that flow rate increases slightly with an increase
in the slope of the bin bottom.
The studies of Reiner (1968) showed that rate of
discharge begins to increase when the angle of inclination
from the bottom of the hopper becomes larger than the angle of
repose of the material. He proposed the optimum angle of
inclination should be in the range of 45• to 60·.
Brown and Richards (1970) have studied the angle of
approach (i.e., the angle of sliding to the vertical, ~v) on
different orifice openings, though not the hopper angle.
Their results show that the angle of approach is practically
independent of the inclination of the side wall when the
inclination of the side wall is less than the angle of sliding
to the horizontal. However, no direct measurements of the
angle of sliding have been made. Their derived results had
22
also shown that the flow rate was a direct function of ~<Pv>
for a slot or y(Pv> for a circular aperture. The functions of
~<Pv> and Y<Pv> were given as:
= (1 - cospv $·sin1
"5 Pv
(2.8)
(2.9)
Davidson and Nedderman (1973) developed prediction
equations of an analytical form based on continuum mechanics
and showed that the flow rate is a function of the reciprocal
of sin9v, where 9v is the hopper angle to the vertical. Their
equation was given as:
(1 + .K) ·g 2·(2·K- 3)
( 2. 10)
where K is an angle function and is given as (l+sin~)/(1-
sin~). However, hopper angle has no significant effect on the
flow rate according to their equation. This work was the
closest to the work described in this research project. The
major differences were the function term used to describe the
effect of wall inclination on the flow rate and the model used
to predict the flow rate without taking into account the
effect of the angle of sliding.
23
Zenz (1976) studied the effect of hopper angle on
granular flow rate from smooth steel bins. He concluded that
flow rate does not depend on wall inclination if the hopper
angle is less than the inclination of the stagnant region
(i.e., the angle of sliding to the horizontal). Zenz (1976)
proposed that flow rate is a function of (tan•) -o · 5 • His
equation was given as:
(2.11)
where Pf is the density of surrounding medium, such as air,
and 4 is the particle void fraction.
Myers and Sellers ( 1977) conducted experiments in a
wedge-shaped hopper. Their results showed that flow rate is
a function of (tan9v>-0.204.
Williams (1977) carried out a series of experiments on a
conical hopper to obtain measured discharge rates over a range
of hopper angles for comparison with the predicted values from
a set of momentum equations. The primary equation of his
model was given as:
(1 + .K) 2·(2·K- 3)
24
(1 - cos1 • 58) sin2 • 58
(2.12)
However, a complete solution of the equations of motion of
granular materials flowing in a conical hopper had not yet
been obtained.
Laird and Roberts (1979), following Zenz (1976), drew the
same conclusion that the flow rate does not depend on hopper
angle if the hopper angle is less than the angle of sliding to
the horizontal. Their equation gave the same function as
Zenz's model and was given as:
M = p (2.13)
where nLR is a coefficient ranging from 0. 29 to 0. 434 for
different materials tested. They also considered cases where
the hopper wall is not of constant slope and found that flow
rate is dependent only on wall angle of the lower section of
the hopper.
2.3 Correlations of Gregory and Fedler
In 1986, Gregory and Fedler developed a mathematical
model based on a balanced force analysis relating downward
forces which cause flow by gravity to upward forces which
resist flow by friction. A comparative study of four orifice
flow models with the same data set was made by Fedler (1988)
and led to the understanding that the Gregory and Fedler
(1986) model is the most reliable for predicting flow of
granular materials through horizontal orifices less than 12 em
25
in diameter and for a wide range of dry types of granular
materials.
In 1987, Gregory and Fedler expanded their original
procedure by developing a method to predict their flow
resistance coefficient in terms of the smallest of the three
dimensions of the granules. The advantage of this expansion
was that the model can be used without extensive and costly
laboratory testing. The Gregory and Fedler (1987) model was,
therefore, chosen to be used in the experimental design for
this work.
As examined by Al-Din and Gunn (1984), the application of
the Fowler and Glastonburg (1959), and the Beverloo et al.
(1961) equations respectively, had shown that the experimental
data for long rectangular orifices were poorly correlated
because the hydraulic diameter used in both formulas was
unaffected by an increase in the slit length. In addition,
the development of Fedler and Gregory (1989a) model was mainly
based on circular orifices. Though their model could be
applied to other orifices, such as a rectangular one, it was
necessary to extend their model to study the flow of granular
materials through rectangular orifices in order to obtain a
better correlation that describes the flow.
In summary, a number of models relating to the effect of
hopper angle on the flow rate have been developed. However,
no experimental or theoretical work has been done to have a
better agreement among these models. It can be said that
26
there is no equation that can be used without any restriction
for all types of hoppers and all kinds of granular materials.
Though the equations were different because of the different
factors they contain, each could be used over a limited range
only. It was therefore necessary to extend the work of Fedler
and Gregory (1989a) to account for the effect of hopper angle
to better describe the flow of bulk solids and powders through
orifices in a hopper device.
27
CHAPTER III
EXPERIMENTAL SETUP AND PROCEDURE
3.1 The Model Hopper
The schematic diagrams of the model hopper setup are
shown in Figure 3 .1. The hopper is 63 em high, 16 em wide and
122 em long. Material of construction of the back wall of the
wedge-shaped hopper device was made of plywood with a
galvanized steel inside facing. The front face of the hopper
was made of transparent plexiglas having the same dimension of
the back wall so that the flow process could be observed from
both the top surface and the front face. Both the front face
and back wall were vertical. They were fixed with their lower
edges to the bottom plate of the slot, which was made of
plywood, by mounting screws through them with a galvanized
steel plate attached to the bottom of the front face.
Sandwiched between the back wall and the window were two
strips of plywood again with galvanized steel inside facings,
61 em long, 15.2 em wide, and 1.3 em thick, equally inclined
to the horizontal to form the inclined walls of the hopper.
The side walls could be set at any inclination to the
horizontal from s• to go• by means of screw threads attached
to them. Care was taken to keep the lower edges of the side
walls parallel and tightly onto the horizontal bottom plate.
It was possible to attach different facing materials to these
28
I
I
I
rectangular m:ifice
top view
•
Orifice
l Hopper Outle:.
to basket
inclim~d wall
plex iglas
•
' \ \
\
Figure 3.1. Front and top views of the model hopper
29
side walls and, also, to change them to different widths. The
hopper was symmetric about the centerline.
In the present work, the exit slot was always a
rectangular orifice with a fixed width of 15.2 em and
changeable length from o to 15. 2 em. Both width and length of
the exit slot were long enough (i.e., larger than 10 em) to
avoid end-wall effects as stated by Brown and Richards (1970).
The length of the exit slot was adjusted by means of screw
threads and measured by means of a ruler. Connected by bolts
and nuts, the hopper was further strengthened by placing an
angled steel bar diagonally across the outside window to
prevent it from protruding into or out of the flow channel,
which would eventually lead to significant variations in the
frictional resistance compared to the surrounding walls.
A stand made of plywood, 130 em long, 64 em wide and 91
em high, was also used to support the hopper device and for
convenient loading of the discharging granular materials from
the exit slot to a receiving basket below the opening. The
hopper device had a hold up capacity of approximately 23 kg of
granular material. The capacity of the hopper was mainly
dependent on the volume, not on the weight of the material
tested.
several different sizes of the orifice opening, of length
2.5 em, 3.8 em, 5.1 em, 7.6 em, 8.9 em, 10.2 em, 12.7 em, 13.3
em and 15.2 em, were chosen for the experiments. Six
30
different side wall angles, so·, 10•, 6o·, so·, 40•, and Jo•,
were also used as needed.
3.2 Tbe Procedure
The mass of the granular material was loaded by pouring
granular solids to the top of the hopper. Flow was initiated
by opening a manually operated sliding gate, made of
galvanized steel. The flow was stopped by inserting the
sliding gate back into the exit slot. Each experimental run
varied from a few seconds for large orifice openings and large
side wall angles to 5 minutes for small orifice openings and
small side wall angles. Since only batch runs were conducted
with an total initial charge of about 20 kg of granular
material, the run time was not very long. It was possible to
prolong the run time to obtain more consistent data by using
a bigger initial charge. This was not done due to the limited
capacity of the hopper and the difficulty of manually
transporting a large quantity of material over a vertical
distance of about 1.6 m. After the completion of each run,
the material collected in the receiving basket was weighed and
returned to the hopper for replication. The mass flow rate of
the material for each run was calculated by dividing the
material weight by the run time.
Five replicated test runs were conducted for each orifice
opening and each side wall angle in order to check
variability. Before and after a series of experiments for
31
each granular material, measurement of moisture content,
relative humidity of the environment, temperature, angles of
sliding on galvanized steel and plexiglas, emptying and
filling angles of repose, particle size range, and material
bulk density were made. Values of the material angle
properties such as angle of sliding on galvanized steel plate
and plexiglas sheet, and emptying and filling angles of repose
for each material tested are listed in Table 3.1.
Moisture content of the material was obtained by weighing
the sample before and after 24 hours of drying in a
conventional utility drying oven at 101•c, and dividing the
difference (i.e., water content) by its dry weight. Relative
humidity was measured using a psychrometer called Psychro-Dyne
model PP100 and readings were read from a slide rule humidity
calculator. Temperature was measured using a thermometer.
The angles of sliding of the material were measured with a
single layer of particles sliding and rolling down a
galvanized steel plate or plexiglas sheet inclined at a
specific angle.
digital caliper.
Particle size was measured by an electronic
Bulk density was measured using a 1 quart
Ohaus test weight apparatus. The container was filled by the
test weight procedure and the measured volume was converted to
cm3. Ideally, the bulk density should be measured during the
flow process, but this was not done due to complexity of the
measurement and lack of suitable equipment.
32
Table 3.1. Values of material angle properties
Material Filling Emptying Type Angle of Angle of
Repose Repose
pp• 18 21
LDPEb 25 27
Nylon 6 40 37
Sorghum 28 40
Flour 44 88
Cornmeal 43 65
LLDPEc: 26 30 ppsd 27 32
•Polypropylene ~ow Density Polyethylene c:Linear Low Density Polyethylene dpolyphenylene Sulfide
33
Angle of Angle of Sliding on Sliding
Steel on Plexiglas
18 23
18 36
22 32
20 25
46 85
36 48
32 36
28 60
In addition, the material flow resistance factor was
measured by the same procedure described above with a
standpipe made of galvanized steel, 58 em high and 25 em in
diameter, seated onto a flat bottom with a circular orifice
opening ranging from 2. 3 em to 10.3 em in the center. Various
orifices sizes were used and four replications were made for
each run. Detailed experimental procedures of the material
resistance factor were discussed by Gregory and Fedler (1986).
Values of the material bulk density and flow resistance factor
for each material tested are listed in Table 3.2.
3.3 The Granular Materials
Basically two types of granular materials were used in
the experiment. First, polymeric pellets such as free flowing
(i.e., non-cohesive) Polypropylene (PP) and Low Density
Polyethylene (LOPE), and bulk solids such as cohesive reground
Nylon 6 and free flowing dry sorghum; second, the polymeric
flakes such as free flowing Linear Low Density Polyethylene
(LLDPE) and Polyphenylene Sulfide (PPS), and powders such as
very cohesive flour and cornmeal. Materials were roughly
categorized according to their angle of repose from
Cheremisinoff (1986). These materials were chosen because
they varied somewhat in bulk density, particle size range and
shape, flow properties, and chemical composition.
34
Table 3.2. Values of material bulk density and flow resistance factor
Material Bulk Density Type p.., (g/cm3)
pp8 0.5344
LDPEb 0.4948
Nylon 6 0.5366
Sorghum 0.6752
Flour 0.5178
Cornmeal 0.5826
LLDPEC 0.4043
ppsd 0.4058
•Polypropylene bLow Density Polyethylene cLinear Low Density Polyethylene dPolyphenylene Sulfide
Flow Resistance Factor K (gjcm2s)
17.6311
15.5092
19.0650
24. 6000e _f
6.6326
12.3424
12.7791
~alue obtained from Gregory and Fedler (1989a) fvalue unable to obtain due to severe arching effect
35
CHAPTER IV
MODEL DEVELOPMENT
The mathematical model previously developed by Gregory
and Fedler (1986) was used to predict the flow of granular
materials through horizontal orifices as a function of orifice
size, bulk density and a flow resistance factor. The material
bulk density was used in the correlations because it is a
combination of particle density and initial packing
characteristics of the material and it reflects the change of
moisture in the environment. Several assumptions were used:
(1) Constant bulk density for each material during
testing,
(2) Straight curvature of the inclined side walls,
(3) Equal wall friction on all sides of the hopper.
The equation for obtaining mass flow rate was (Gregory
and Fedler, 1986):
A2 2 M = g·~· Pb
p C K
where,
MP = p~edicted mass flow rate, qjs
q = gravitational constant, cmjs2
Ac = cross sectional area of the orifice, cm2
c = perimeter of the orifice, em
pb = material bulk density, qjcm3
K = resistance coefficient, qjcm2s.
36
( 4. 1)
The resistance coefficient, K, (or material resistance
factor) in equation ( 4 .1) was a combination of material
factors such as surface roughness and particle size (Fedler
and Gregory, 1987). It could be described as follows:
(4.2)
where,
~1 = 10.41 gjcm2s for Type I materials (particles
with smooth surface and regular shape)
= 5.45 gjcm2s for Type II materials (particles
with rough surface and irregular shape)
r. = minimum length of the particle, em.
The detailed development of individual components of the
above prediction equations was discussed by Fedler and Gregory
(1986), Gregory and Fedler (1987), Fedler and Gregory (1988),
and Fedler and Gregory (1989a). Based on these prediction
equations, a complete model for both horizontal and non
horizontal orifices had been presented and discussed by
Gregory and Fedler (1989b). However, a slight deviation
existed between the predicted and experimentally measured mass
flow rates when equation (4.1) was applied to hopper with two
inclined side walls. A simple correlation was used to
eliminate the deviation of equation (4.1).
(4.3)
37
where,
At = adjustment factor
~ - experimentally measured mass flow rate, gjs.
Based on the theory of empty annulus (Beverloo et al.,
1961 and Brown and Richards, 1970), the horizontal force of
each particle was eliminated when they collapsed with each
other around the orifice opening during flow and only the
vertical force of each particle remains. Thus, there was an
energy loss upon particle impact as stated by Drake (1986).
Correlations for the adjustment factor were described as
follows:
Af = 1. 0 when 6 < (} (4.4)
( 4. 5)
where,
e = the hopper side wall angle to the horizontal
13 = the angle of sliding of the material to the
horizontal on a galvanized steel plate
A, = coefficient
~ - coefficient.
The function (1 - cos6) have also been used directly in
the correlations of Brown and Richards (1970) and that of
Williams (1977) with additional power term, and indirectly in
38
the correlation of Davidson and Nedderman (1973) where
( \sin26) • ( 1 - cos6) •
Regression analysis of the data was performed using a
program called MERV (Gregory and Fedler, 1986). Values of
coefficients A1 and ~ at various orifice sizes for each
material were obtained by regression analysis of the
experimental data and are listed in Table 4.1 and 4.2.
The coefficient ~ for most materials tested appeared to
be a invariant with the orifice size as shown in Figure 4.1.
Therefore, it was assumed to be an universal constant for all
the materials tested. Its average value was calculated to be
0.4890. Notice that the value of coefficient A2 is
approximately 0.5 for two inclined side walls, whereas it is
1.0 for conical hopper (i.e., four inclined side walls) as
shown in the correlation of Brown and Richards {1970),
Davidson and Nedderman (1973), and Williams (1977).
Therefore, the data indicated that coefficient A2 is
determined by the number of inclined side wall.
On the other hand, as shown in Figure 4.2, the
coefficient A1/K for most materials tested decreased rapidly
over the range of small orifice opening sizes and gradually
approached a constant over the range of large orifice opening
sizes. Correlation of A1 divided by the material resistance
factor, K, of each material tested was assumed to be a
function of orifice opening size. It was assumed to be the
universal relationship for all the materials tested.
39
Table 4.1. Coefficient A1 of each material used in equations (4.5) and (4.6)
Orifice ppe LDPEb Sorghum Opening
Width (em)
2.5 2.3106 1.9012
3.8 2.0610 -5.1 1.8665 1.6395
7.6 1.8096 1.4373
8.9 1.5911 -10.2 1.6015 1.3576
12.7 1.6141 -13.3 - 1.3638
15.2 1.4387 -average 1.7933 1.5371
•Polypropylene bLow Density Polyethylene cLinear Low Density Polyethylene dpolyphylene Sulfide
40
3.5158
-2.4912
2.1221
-1.9772
1.8778
-1.5905
2.2119
LLDPEC ppgd
1.6286 1.6996
- -1.2866 1.6523
0.9833 1.6116
- -1.0618 1.5602
0.9974 1.3931
- -0.9450 1.2719
1.1524 1.6996
Table 4.2. Coefficient A2 of each material used in equations (4.5) and (4.6)
Orifice pp• LDPEb Sorghum Opening
Width (em)
2.5 0.5107 0.5504
3.8 0.5096 -5.1 0.5088 0.4942
7.6 0.5517 0.4012
8.9 0.4941 -10.2 0.5026 0.4209
12.7 0.5103 -13.3 - 0.3955
15.2 0.4527 -average 0.5283 0.4345
•Polypropylene bLow Density Polyethylene cLinear Low Density Polyethylene dPolyphylene Sulfide
41
0.5304
-0.5179
0.4681
-0.5173
0.5256
-0.4114
0.4961
LLDPEC ppsd
0.6112 0.4813
- -0.5412 0.5524
0.3495 0.5686
- -0.5044 0.5594
0.4721 0.4303
- -0.4061 0.4101
0.4989 0.4813
~ "'
4.0
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0.5~--------------------------------~
0.4
0.3
~ ' .-
4
< 0
.2
0.1
~
1 •
R
0 I
u
8 I
o------------~--~~--~--~--~----
0 2
4 6
8 10
12
14
16
O
RIF
ICE
OPE
NING
(e
m)
Fig
ure
4
.2.
Plo
t o
f A
l/K
o
f L
OP
E,
PP
, L
LD
PE
, P
PS
, an
d
dry
so
rgh
um
as
a fu
ncti
on
o
f o
rif
ice
op
en
ing
where,
C1 = Coefficient
C2 = Coefficient
c3 = Coefficient
X = orifice opening lenqth, em.
( 4 0 6)
Using MERV, coefficients c1 , c2 , and c3 of equation (4.6) were
calculated to be 0.8351, 0.0884 and 1.0887 respectively.
By combining equation (4.1), (4.3), (4.5) and (4.6)
together, a modified equation for predicting the flow rate of
granular materials through orifices in a hopper was given as:
It can be seen from this simple equation that the material
resistance factor, x, had been cancelled out. Therefore
recalibration of the coefficients of this model is not
necessary when testing a new material. As a result, this
model can be applied easily and directly to different non-
cohesive granular materials.
44
CHAPTER V
RESULTS AND DISCUSSION
As discussed in the preceding chapter, a model based on
the theory developed by Gregory and Fedler (1986) has been
developed to examine the flow of granular materials through
horizontal orifices in a hopper. This model incorporates the
effect of the hopper angle on the flow rate of both cohesive
and non-cohesive granular materials.
Various flow phenomena have been observed and studied
during the experimentation. Observations of the flow behavior
will be presented in the first section of this chapter.
Results of the test runs on different granular materials will
be presented under two categories, namely bulk solids and
powders.
5.1 Visual Observation of Flow Phenomena
The flow pattern was observed to be asymmetric, generally
for all operational conditions and for all the materials
tested. This reconfirmed the observation of asymmetric flow
patterns by Lee et al. (1974).
The height of the stagnant region at smaller hopper
angles was observed to have no clear connection with the size
of the orifice. This disagrees with the hypothesis of some
workers who have proposed a relationship between height of the
stagnant region and orifice diameter (Novosad and Surapati,
45
1968; Chatlynne and Resnick, 1973). In this work, the height
of the stagnant region was observed to depend on the angle of
sliding of the material on the surface of the hopper wall. A
stagnant region was observed only when the hopper angle was
equal to or less than the angle of sliding. It is possible
that when both angles are equal some material adhesion on the
side wall could still occur. The height of the stagnant
region increases when the hopper angle decreases.
The angle of sliding is a unique material property and an
important parameter in the model. It provides a measure of
the relative adhesiveness of a granular particle to a
dissimilar surface. In this work, during the measurement of
such a property for a given material, the angle fluctuated
between an upper and lower limit for some materials. Usually
the flow took place at the upper limit, but sometimes the
material adhered to the surface of the plate and the flow was
partially inhibited. This reconfirmed the observation of
Brown and Richards (1970) who called the angle of sliding the
angle of approach. The mean values of the angle of sliding
for various materials used in the experiment are listed in
Table 3.1.
The angle of sliding was observed to be independent of
the hopper angle. It was noted that the angle of sliding
depends on the type of granular material, the surface
properties of the particles (e.g., surface roughness), the
manner in which materials were placed on the surface of the
46
wall, the rate of change of the angle of the surface during
measurements, the geometry of the hopper, and the aperture
through which the material is flowing. However, no standard
testing method is available in the literature.
Particles in the lower region of the hopper close to the
hopper side walls may be the first ones to move toward the
orifice. During flow, a particle may continually change its
position from the upper region facing the open air to the
lower region and vice versa. This partial-slipping phenomena
of the particles along the hopper side walls was observed
frequently, which reconfirmed the study of Ding and Gidaspow
(1990).
5.2 Bulk Solids
Typical polymeric pellets with a regular shape and smooth
particle surface, such as PP, flowed freely out of the hopper
and exhibited a uniform flow patterns. Very little cohesion
among particles and adhesion of particles onto to the hopper
walls was observed during the flow.
The mass flow rate of PP at each orifice size as a
function of the hopper angle is illustrated in Figure 5.1.
From this figure, it is obvious that the larger the hopper
angle, the higher the mass flow rate. The rate of increases
of flow rate appears to be higher for large hopper angles
(i.e., 60• and larger). In addition, the mass flow rate
increases as the orifice size increases generally in equal
47
,...., r>
" (Jl 'V
,...., w Ul 1-\J <{ ( !r 0
Ul ~ )
0 0 ...J£ U..l-
'V
(f) (f) <{
2
18
17 POLYPROPYLENE
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
0 20 40 60 80
HOPPER ANGLE
0 2.5 em + 3.8 em o 5.1 em ~ 7.;, em X 8.9 em 'V 10.2 <m
0 12.7 em + 15.2 em
Figure 5.1. Plot of mass flow rates of PP as a function of hopper angle at orifice opening of 2.5 to 15.2 em
48
increments. The rate of increase of flow rate tends to be
higher for large orifice sizes. These results show that the
mass flow rate increases more rapidly for large hopper angles
and orifice sizes. As a result, the mass flow rate of PP is
clearly a function of both hopper angle and orifice size.
The average adjustment factor for PP as a function of
hopper angle, without considering the effect of orifice size,
is shown in Figure 5.2. Equations (4.4) and (4.5) accurately
estimate the values of the average adjustment factor for PP.
The adjustment factor for PP at each orifice size as a
function of the hopper angle is illustrated in Figure 5.3.
The adjustment factor increases as the hopper angle increases
or orifice size decreases. From this figure, it is clearly
indicated that the adjustment factor is also a function of
orifice size. Table 5.1 lists the values of adjustment factor
for PP at various hopper angles. From this table, it shows
that the original model of Gregory and Fedler (1987)
underestimates the flow rate of PP almost entirely for hopper
with two inclined side walls. This disagrees with the
hypothesis of Kotchanova (1970) that most of the granular flow
models tend to overestimate the actual flow rate of bulk solid
materials.
A plot of the predicted versus measured mass flow rate of
pp is presented in Figure 5. 4. It shows that the model
accurately predicts the flow rate of PP and, therefore, the
offset of the Gregory and Fedler (1987) model is eliminated.
49
ln
0
3.0----------------------------------~
2.5
a:
0 t-
- u 2
.0
<(
LL
I 0
1-- z
1.5
U
J ~
t--
(I)
:::J
1.0
J a <
(
0.5
o~--~--~--~--~--~--~--~--~--~
0 1
0
20
30
40
5
0
60
7
0
ANGL
E FR
OM
HO
RIZO
NTA
L 8
0
90
Fig
ure
5
.2.
Pl8
t o
f av
era
ge
ad
justm
en
t fa
cto
rs
of
PP
as
a fu
ncti
on
o
f h
op
per
an
gle
U'l ~
3.0
. +
Z
.5 c
a M
3
.8 C
111
~ 2
.5 r
P
5,.
! em
o
7.6
em
o
8.9
cm
,10
.2
em
,12
.7
em
+
+
15
.2 e
m
tJ 2
. 0
<!
lJ..
t- z 1
.5
w
::::£
I- en
:::> 1
.0
""")
+
0
I
<t:
0.5
o~--~--~--~--~--~--~--~--~--~
0 10
2
0
Fig
ure
5
.3.
30
4
0
50
6
0
70
8
0
90
ANGL
E FR
OM
HO
RIZO
NTA
L P
lot
of
ad
just
men
t fa
cto
rs
of
PP
as
a fu
ncti
on
o
f h
op
per
an
gle
at
orif
ice
op
en
ing
o
f 2
.5 to
1
5.2
em
Table 5.1. Values of adjustment factor for polypropylene at various hopper angles
Orifice 3o· 40° so· 60° 70° Opening
(em)
2.5 1.485 1.322 1.343 1.585 2.191
3.8 1.258 1.206 1.260 1.469 1.879
5.1 1.119 1.127 1.178 1.311 1.644
7.6 0.921 1.032 1.157 1.336 1.540
8.9 0.945 0.990 1.054 1.159 1.349
10.2 - 1.014 1.055 1.173 1.354
12.7 - 0.998 1.071 1.176 1.363
15.2 - - 1.051 1.089 1.193
average 0.967 1.063 1.146 1.287 1.564
52
"" lll '\. 01 '"" ~ i-( ''"' IV II) -"0
~ c 0
C L~ ....: J L 0
0~ ~"" ' ., '-' Q u :--
20r-~------------====================~
7
6
5
4
3
2
0 0
POLYPROPYLENE
R2=.9727
2 4
r/
~J co
0
5 8
0
10 (Tho ... ~c- :: c 3)
12
MU.SURED FLOW RA.IT (g/s)
5.4. Plot of predicted versus mass flow rate of pp
53
0
0
0
14 16 18
measured
The raw data of mass flow rate of PP is presented in Table
A.l.
Materials, such as LOPE, show strong cohesion among
particles and strong adhesion of particles onto the hopper
walls, primarily due to electrostatic effects. The cohesive
forces are mainly Van der Waals forces. They increase as
particle size is reduced and also vary significantly with
relative humidity. However, the precise mechanism as well as
the nature of the electrostatic charge on solid surfaces are
not completely understood.
The mass flow rate of LOPE at each orifice size as a
function of the hopper angle is illustrated in Figure 5.5.
The mass flow rate of LOPE increases as the hopper angle and
orifice size increase. The rate of increase is higher for
large hopper angles and orifice sizes. The mass flow rate of
LOPE is, therefore, a function of both hopper angle and
orifice size. In addition, the mass flow rate of LOPE is
shown to be smaller than that of PP at a fixed orifice due to
strong cohesion caused by electrostatic charges.
The average adjustment factor for LOPE as a function of
hopper angle, without considering the effect of orifice size,
is shown in Figure 5.6. Equations (4.4) and (4.5) accurately
estimate the values of the average adjustment factor for LOPE.
The adjustment factor for LOPE at each orifice size as a
function of the hopper angle is illustrated in Figure 5.7.
The adjustment factor increases as the hopper angle increases
54
"' ., " ()I 'V'
'"' w Ul 1-"0 ( c lr 0
~ ~ 0 0 _J.( 11.1-
'V'
U) U) ( ~
14
13 LOW DENSITY POLYETHYLENE
12
11
10
9
8
7
6
5
4
3
2
0 20 40 60 80
HOPPER .;NG ~:::
0 2.5 em + 5.1 cr:1 0 7.6 em ~ 10.2 em X • 3.3 em
Figure 5.5. Plot of mass flow rates of LOPE as a function of hopper angle at orifice opening of 2.5 to 15.2 em
55
ln
0'1
3.0----------------------------------~
2.5
cr
0 J- u
2.0
<
( LL
.... z 1
.5
lLI ~ ._ ~ 1
.0 L
0
, 0 <(
0.5
o~~~~~~--~--~--~--~--~--~
0 1
0
20
30
40
50
60
70
ANGL
E FR
OM
HO
RIZO
NTA
L 8
0
90
Plo
t o
f av
era
ge
ad
just
men
t fa
cto
rs
of
LO
PE
as
a fu
ncti
on
o
f h
op
per
an
gle
F
igu
re 5
.6.
0'1
..J
3.0
I 0
2.5
em
2
.5
+
5.1
em
a:
*
7.6
em
0 .....
u
2.0
tl
1
0.2
em
<
t 0
13
.3
em
0 lL
t-- z
1.5
U
J ~
..__
en
:::J
1.0
J a <
!
0.5
0----~--~--~--~--~--~--~--~~
0 1
0
20
30
40
50
60
70
ANGL
E FR
OM
HO
RIZO
NTA
L 80
90
Fig
ure
5
.7.
Plo
t o
f ad
justm
en
t fa
cto
rs
of
LO
PE
as
a fu
ncti
on
o
f h
op
per
an
gle
at
orif
ice
op
en
ing
o
f 2
.5 to
1
5.2
er
n
or orifice size decreases. From this figure, it is clearly
indicated that the adjustment factor is also a function of
orifice size. Table 5. 2 lists values of adjustment factor for
LOPE at various hopper angles. From this table, it shows that
the original model of Gregory and Fedler (1987) almost
entirely underestimates the flow rate of LOPE for hopper with
two inclined side walls.
A plot of the predicted versus measured mass flow rate of
LOPE is presented in Figure 5.8. It shows that the model
accurately predicts the flow rate of PP and largely eliminates
the offset of the Gregory and Fedler (1987) model. The raw
data of mass flow rate of LOPE is presented in Table A.2.
In the case of reground Nylon 6, very turbulent flow
behavior occurs. This exceptional behavior of reground Nylon
6 is probably due to its unique particle structure, limited
amount of testing material available, shorter time
measurements, and thus larger operational errors. Materials
with irregularly shaped particles and varying particle sizes
tend to show undesirable segregation and irregular flow
patterns due to strong cohesion among the particles. Strong
inter-particle cohesive forces during the flow was apparent
due to the strong ability of the particles to absorb moisture
from the surrounding environment.
The mass flow rate of reground Nylon 6 at each orifice
size as a function of the hopper angle is illustrated in
Figure 5.9. The mass flow rate of reground Nylon 6 increases
58
Table 5.2. Values of adjustment factor for low density polyethylene at various hopper angles
Orifice 40° so· 60° 1o· Opening
(em)
2.5 1.128 1.151 1.303 1.735
5.1 1.073 1.084 1.171 1.400
7.6 1.033 1.039 1.099 1.281
10.2 0.925 0.960 1.068 1.213
13.3 0.973 1.010 1.074 1.171
average 1.022 1.043 1.125 1.320
59
0 2 4 6 8 (Thousands)
MEASURED FLOW RATE (g/s)
10 12
5.8. Plot of predicted versus measured mass flow rate of LOPE
60
""' 0
" 01 v
""' w 0 1-'0 <{ c 0:: 0
0 ~ J 0 0 _J.( ~~--
v
Vl Vl <(
~
15
14 REGROUND N)l CJN 6 / 13
12 /
I 11
10 /~
9
8 ~-
7
6
5
4
3
2
0 20 40 60 80
HOPPER ANG~:
0 2.5 em + 5.1cm o 7.6cm t:. 1CZ cm X 12.7cm 'il 15.2cm
Figure 5.9. Plot of mass flow rates of reground Nylon 6 as a function of hopper angle at orifice opening of 2.5 to 15.2 em
61
as the hopper angle or orifice size increase. The rate of
increase is higher for large hopper angles and orifice sizes.
Thus, the mass flow rate of reground Nylon 6 is clearly a
function of both hopper angle and orifice size. In addition,
the mass flow rate of reground Nylon 6 is smaller than that of
PP and LOPE at a fixed orifice size due to much stronger
cohesion caused by the nature of the particles.
The average adjustment factor for reground Nylon 6 as a
function of hopper angle, without considering the effect of
orifice size, is shown in Figure 5.10. Equations (4.4) and
(4.5) accurately estimate the values of the average adjustment
factor for reground Nylon 6. The adjustment factor for
reground Nylon 6 at each orifice size as a function of the
hopper angle is illustrated in Figure 5 .11. From this figure,
there appears to be no specific trend with respect to the size
of the orifice and, therefore, the adjustment factor for
reground Nylon 6 is not a function of orifice size. Table 5. 3
lists the values of adjustment factor for reground Nylon 6 at
various hopper angles. From this table, it shows that the
original model of Gregory and Fedler (1987) underestimates the
flow rate of reground Nylon 6 at large hopper angles (i.e.,
60• and larger) and orifice size of 10.2 em for hopper with
two inclined side walls.
A plot of the predicted versus measured mass flow rate of
reground Nylon 6 is presented in Figure 5.12. It shows that
the model prediction and the experimental measurement are in
62
0\ w
((
0 1-
3.0
~----------------------------
2.5
u 2
.0
<t
lL
1--
-~
.,._;_
l.LJ
2 1--- r,,
----, - -, 0 <.(
a
1 ~
1 ll
c
_._
_..--
--.
-------
-----
'
~---u
--_
o --
----
cr· --
-----
-cJ'
0.
5 --
\_)
__
_ L
___ _
_
_ _
!_ _
__
_ ..
....
.L_
_ -
· --
--L
---
0 .1
'
.l.v
2
J :3-
~~
/ ~ :;
: .. C·
~~
~l
-~
~:.( r
, 90
Fig
ure
5
.10
.
AN
GLE
r R
iJ ~-t~
' ~
L 11 n
z J ~ J
r A L
Plo
t o
f av
era
ge
ad
just
men
t fa
cto
rs
of r~ground
Ny
lon
6
as
a fu
ncti
on
o
f h
op
per
an
gle
0----~------~--~------~--~--~--~
0 10
20
30
40
50
60
70
AN
GLE
FROM
H
ORI
ZON
TAL
80
90
Fig
ure
5
.11
. P
lot
of
ad
just
men
t fa
cto
rs
of
reg
rou
nd
N
ylo
n
6 as
a fu
ncti
on
o
f h
op
per
an
gle
at
orif
ice
op
en
ing
o
f 2
.5 to
1
5.2
em
Table 5.3. Values of adjustment factor for reground nylon 6 at various hopper angles
Orifice 3o· 40° so· 60° 1o· Opening
(em)
2.5 0.884 0.840 0.865 1.100 -5.1 0.914 0.882 0.958 1.105 1.446
7.6 0.944 0.912 0.987 1.117 1.398
10.2 - 1.098 1.187 1.303 1.445
12.7 - 0.913 0.993 1.246 -15.2 - 0.916 0.967 1.089 -
average 0.889 0.927 0.993 1.160 1.476
65
18
17 REGROUND NYLON 6 0
16 R2=.9569
15 0
14 0
,...., 13 ~
" 01 12 '-../
w 0 ~,...., 11 ( Ill 0 !r"' 10 ~ c 0 ~ 9 ...J ) ll.. 0
8 0~ W'V
7 ~ 0 0 6 w 0 !r (L 5
4
3 0 ,..J >='
2
f 0
0 2 -! 6 8 10 12 14 (Th0usonds)
MEASURED FlOW RATE (g/s)
5.12. Plot of predicted versus measured mass flow rate of reground Nylon 6
66
good agreement. Our work largely eliminates the offset of the
Gregory and Fedler (1987) model except under high flow rate
condition. The raw data of mass flow rate of reground Nylon
6 is presented in Table A.J.
Because of the nature of the grain material, sorghum
tends to be hygroscopic and absorb large amounts of moisture
than PP and LOPE. Thus, the material was preconditioned by
storing in a dry atmosphere before the start of an
experimental run. A higher flow rate of dry sorghum was
obtained compared to that of other bulk solids tested, which
confirmed the results of Chang and Converse ( 1988) . A
possible explanation, proposed by Chang and Converse (1988),
is that the fine hair on kernel surfaces becomes softer at
high moisture conditions prior to drying. The softer hair may
help reduce the friction between kernels during movement,
which results in higher flow rates.
The mass flow rate of dry sorghum at each orifice size as
a function of the hopper angle is illustrated in Figure 5.13.
The mass flow rate of dry sorghum increases as the hopper
angle and orifice size increase. The rate of increase is
higher for large hopper angles and orifice size. The mass flow
rate of dry sorghum is, therefore, a function of both hopper
angle and orifice size. In addition, the mass flow rate of
dry sorghum is shown to be the largest among that of other
bulk solid materials such as PP, LOPE and reground Nylon 6, at
a fixed orifice size.
67
,..., til
" (Jl ....., ,...,
w til 1-"0 ~ ( oc 0
• VJ ~ )
0 0 _.J.( t:_l-....., ~') Vl ~ 2
22
DRY SORGHUM 20
18
16
14
12
10
8
6
4
2
0 0 20 40 60 80
HOPPER h ~ <G LE
D 2.5 em + 5.1 em 0 7.6 em 6 10.2 em X 12.7 em 'iJ 15.2 em
Figure 5.13. Plot of mass flow rates of dry sorghum as a function of hopper angle at orifice opening of 2.5 to 15.2 em
68
The average adjustment factor for dry sorghum as a
function of hopper angle, without considering the effect of
orifice size, is shown in Figure 5.14. Equations (4.4) and
(4.5) accurately estimate the values of the average adjustment
factor for dry sorghum. The adjustment factor for dry sorghum
at each orifice size as a function of the hopper angle is
illustrated in Figure 5.15. The adjustment factor increases
as the hopper angle increases or orifice size decrease. From
this figure, it is clearly indicated that the adjustment
factor is also a function of orifice size. Table 5.4 lists
the values of adjustment factor for dry sorghum at various
hopper angles. From this table, it shows that the original
model of Gregory and Fedler (1987) entirely underestimates the
flow rate of dry sorghum for hopper with two inclined side
walls.
A plot of the predicted versus measured mass flow rate of
dry sorghum is presented in Figure 5.16. It shows that the
model prediction and the experimental measurement are in good
agreement only at low flow rate condition. Therefore, the
offset of the Gregory and Fedler (1987) model remains under
high flow rate condition. The raw data of mass flow rate of
dry sorghum is presented in Table A.4.
The coefficient A1 in equations (4.5) and (4.6) for PP,
LOPE, reground Nylon 6 and dry sorghum as a function of
orifice size is illustrated in Figures 5.17 to 5.20. The
coefficient A1 seems to be a superimposed exponential and
69
-..J
0
3.0~-----------------------------------
2.5
cr
0 .....
. u
2.0
<
t lL
...... z
1.5
lJ
J ::E
.....
. (/
) ::>
1
. 0
J 0 <t
0.5
o~--~--~--~--~--~--~--~--~--~
0 10
20
3
0
40
50
6
0
70
8
0
90
ANGL
E FR
OM
HO
RIZO
NTA
L F
igu
re
5.1
4.
Plo
t o
f av
era
ge
ad
just
men
t fa
cto
rs
of
dry
so
rgh
um
as
a fu
ncti
on
o
f h
op
per
an
gle
'-1
.....
3.0
I 0 -
2.5
a:
0 t- u
2.0
<
{ lJ.
..
1-- z
1.5
w
~ .._ ~ 1
.0 t
::::-u=
0
o 2
. 5
em
+
5.1
em
*
7. 6
em
t2
10
.2 e
m
0.5
I o
12
.7 e
m
J
0 '
I I
L
I I
I 0
1~ • 2
em
,
0 10
2
0
30
4
0
50
6
0
70
8
0
90
Fig
ure
5
.15
.
ANGL
E FR
OM
HO
RIZO
NTA
L P
lot
of
ad
just
men
t fa
cto
rs
of
dry
so
rgh
um
as
a fu
ncti
on
o
f h
op
per
an
gle
a~ o
rif
ice
op
en
ing
o
f 2
.5 to
1
5.2
em
Table 5.4. Values of adjustment factor for dry sorghum at various hopper angles
Orifice 3o· 40° so· 60° 70° Opening
(em)
2.5 - 2.022 2.333 2.664 -5.1 - 1.475 1.653 1.900 -7.6 1.337 1.378 1.484 1.471 1.780
10.2 1.157 1.177 1.235 1.415 1.671
12.7 1.062 1.136 1.193 1.359 1.474
15.2 1.055 1.077 1.161 1.281 1.326
average 1.287 1.378 1.462 1.682 1.816
72
32
30 DRY SORGHUM
R2=.8786 0
28
26 0
'"" 24 0
Ul
" 22 oD c;. '- '
_; 2C· oo ,,-,
' Ul 16 0 r'O ~ 6 0 0 Ul 1E
0 _! ) 0 :..... . 0
(' 14 0 .,-- f-~'-"'
r- 12 :...l ,:) !.:..J .v 1C '-~ - 8
6 0
0 t
2
0 3 5 7 9 11 13 15 17 19 21
(Tho LtScrds) MEASURED FLOW PATE (g/s)
5.16. Plot of predicted versus measured mass flow rate of dry sorghum
73
4.0
3.5
3.0
2.5
-.J
~ 2
.0
,e.
1.5
1.0
0.5
0
0 0
L-------~--------~--------L-----
---l
I 1
0 2
4 6
8 1
0
12
14
O
RIF
ICE
OPE
NING
(e
m)
_Fig
ure
5
.17
. P
lot
of
A1
co
eff
icie
nt
of
PP
as
a fu
ncti
on
o
f o
rif
ice
op
en
ing
siz
e
16
4.0
3.5
3.0
2.5
...,J
~ 2
.0
Ul
I
1.5
1.0
0.5
0
~
0 2
Fig
ure
5
.18
.
u ,
0
l __
....._
_ __
__
.__
.
4 6
8 O
RIF
ICE
OPE
NING
10
(e
m)
12
1
4
Plo
t o
f A
1 co
eff
icie
nt
of
LO
PE
as
a fu
ncti
on
o
f o
rif
ice
op
en
ing
siz
e
16
4.0
3.5
3.0
2.5
.....]
~ 2
.0
0\
I
1. 5
1. 0
0.5
0 0
0
.,--
0
2 4
6 8
10
12
14
O
RIF
ICE
O
PEN
ING
(e
m)
Fig
ure
5
.19
. P
lot
of
A1
co
eff
icie
nt
of
reg
rou
d
Ny
lon
6
as
a fu
ncti
on
o
f o
rifi
ce
op
en
ing
siz
e
16
4.0
3.5
0
3.0
2.5
'-l
f~ 2
.0
'-l
1. 5
1.0
0.5
0 0
2
Fig
ure
5
.20
.
..1
• __
_ ....
4 6
8 10
12
14
16
O
RIF
ICE
OPE
NING
(e
m)
Plo
t o
f A
1 co
eff
icie
nt
of
dry
so
rgh
um
as
a fu
ncti
on
o
f o
rif
ice
op
en
ing
siz
e
linear function for the wide range of orifice sizes tested for
all bulk solid materials except reground Nylon 6. Moreover,
The coefficient ~ in equation (4.5) for PP, LOPE, reground
Nylon 6 and dry sorghum as a function of orifice size is
illustrated in Figures 5.21 to 5.24. The coefficient A2
appears to be a constant for each bulk solid material.
However, the values of coefficient~ for reground Nylon 6 are
much lower than those obtained for other materials.
Coefficients A1 and ~ for PP, LOPE, and dry sorghum as a
function of orifice size are shown in Figures 4.1 and 4.2.
5.3 Powders
Fine powders, such as flour and cornmeal, tend not to be
free-flowing materials (i.e., cohesive materials). Small
particles tend to aggregate together due to strong inter
particle cohesive forces and strong adhesion to the hopper
walls made. Thus, the flow of these granular materials were
very slow and irregular. A severe arching (i.e. , a dome
formed just above the orifice) or ratholing (i.e., large
stagnant region around the orifice and along the hopper walls)
occurred very often during the flow of flour and cornmeal.
Complete stoppage of the flow occurred even in the case of
large orifice size, especially for a very fine powdery
material such as flour. As a result, experimental values of
material resistance factor, and, consequently, values of
adjustment factor for flour were difficult to obtain. Results
78
4.0
3.5
3.0
2.5
....]
~ 2
.0
\0
1. 5
1. 0
0.5
0 0
.. ..
J •--
2 4
6 8
10
12
1
4
OR
IFIC
E O
PENI
NG
(em
)
Fig
ure
5
.21
. P
)ot
of
A2
co
eff
icie
nt
of
PP
as
a fu
ncti
on
o
f o
rifi
ce
op
en
ing
siz
e
16
0)
0
4.0~--------------------------------~
3.5
3.0
2.5
~ 2
.0
1.5
1.0
0.5
a
-b
•
a
0 ~
0 2
Ll 6
8 10
12
O
RIF
ICE
OPE
NIN
G
(em
) 14
1
6
Fig
ure
5
.22
. P
lot
of
A2
co
eff
icie
nt
of
LO
PE
as
a f~nction
of
ori
fice o
pen
ing
siz
e
())
~
'
4.0
3.5
3.0
2.5
~ 2
.0
1.5
1.0
0.5
oL_._
0
2
o "
e e
a e
---J'-
-------L
----..
....
_---'-
--
··--....
.l·----1
--.-
4 6
8 O
RIF
ICE
OPE
NING
10 (em
) 1
2
14
1
6
Fig
ure
5
.23
. P
lot
of
A2
co
eff
icie
nt
of
reg
rou
nd
N
ylo
n
6 as
a fu
ncti
on
o
f o
rif
ice
op
en
ing
siz
e
4.0
3.5
3.0
2.5
0)
~ 2
.0
lV
I
1.5
1.0
0.5
0
• •
a Q
0
b
0 2
Fig
ure
5
.24
.
4 6
8 O
RIF
ICE
OPE
NING
10 (em
)
...! _
__
_. _
__
_
12
1
4
16
Plo
t o
f A
2 co
eff
icie
nt
of
dry
so
rgh
um
as
a fu
ncti
on
o
f o
rif
ice o
pen
ing
siz
e
were fortunately obtained for cornmeal. However, the accuracy
of the data is questionable due to the irregular flow pattern.
The mass flow rate of flour at each orifice size as a
function of the hopper angle is illustrated in Figure 5.25.
The mass flow rate of flour increases as the hopper angle and
orifice size increase. The rate of increase is higher for
large orifice sizes. Thus, the mass flow rate of flour is
clearly a function of hopper angle and orifice size. It was
also noted that larger stagnant regions occurred for low
hopper angles (i.e., less than so•) and arching occurred at
smaller orifice sizes (i.e. , smaller than 8. 9 em) . In
addition, the mass flow rate of flour is much smaller than
that of bulk solids investigated at a fixed orifice size due
to the nature of the particles. The raw data of mass flow
rate of flour is presented in Table A.5.
The mass flow rate of cornmeal at each orifice size as a
function of the hopper angle is illustrated in Figure 5.26.
The mass flow rate of cornmeal increases as the hopper angle
and orifice size increase. The rate of increase is higher for
large hopper angles and orifice sizes. The mass flow rate of
cornmeal is, therefore, a function of both hopper angle and
orifice size. In addition, the mass flow rate of cornmeal is
shown to be smaller than that of bulk solids investigated and
larger than that of flour at a fixed orifice size.
The average adjustment factors of cornmeal as a function
of hopper angle, without considering the effect of orifice
83
" Ill 01
'-"
" wIll 1-iJ <{ ( a: 0
Ill ~ J c 0 _J..( ~ ....
'-"
lfl lfl <{
~
10
FLOUR 9
8
7
6
5
4
3
2
0 20 40 60 80
HOPPER 1-'i(·L:
0 8.9 em + 10.2 em o ·,:.7 em 6. 15.2 em
Figure 5.25. Plot of mass flow rates of flour as a function of hopper angle at orifice opening of 8.9 to 15.2 ern
84
r. fl)
'\. 01 '-'
r. w II) 1- U ( (
0::: 0 II)
~ J 0 0 _J.( !._I-
v
VJ VJ (
2
15
14 CORNM[~ /
13
12
11
10
9
8
7
6
5
4
3
2
0 2C' 40 60 80
HOPPER ANG~E
0 2.5 em + 5.1 em o 7.6 em 6 10.2 em X 12.7 em 'il 15.2 ern
Figure 5.26. Plot of mass flow rates of cornmeal as a function of hopper angle at orifice opening of 2.5 to 15.2 em
85
size, is shown in Fiqure 5. 27. Equations (4.4) and (4.5)
accurately estimate the values of the average adjustment
factor for cornmeal. The adjustment factor for cornmeal at
each orifice size as a function of the hopper angle is
illustrated in Fiqure 5.28. There appears to be no specific
trend with respect to the size of the orifice. Therefore, the
adjustment factor for cornmeal is not a function of orifice
size. Table 5.5 lists the values of adjustment factor for
cornmeal at various hopper angles. From this table, it shows
that the original model of Gregory and Fedler (1987) entirely
overestimates the flow rate of cornmeal for hopper with two
inclined side walls.
A plot of the predicted versus measured mass flow rate of
cornmeal is presented in Fiqure 5. 29. It shows that the model
does not accurately predict the flow rate of cornmeal and the
large offset of the Gregory and Fedler (1987) model still
remains. The raw data of mass flow rate of cornmeal is
presented in Table A.6.
Flake materials, such as LLDPE, tend to flow freely
through orifices in a hopper with inclined walls. Though
their particle sizes were small, neither arching, ratholing
nor stoppage of flow occurred. However, cohesion among the
particles and adhesion of particles onto the hopper walls was
observed.
The mass flow rate of LLDPE at each orifice size as a
function of the hopper angle is illustrated in Fiqure 5.30.
86
0)
-..J
2.0~--------------------------------~
~ 1
. 5 I
t- u
..:
LL !z i.
0 .
__ _
__
__
__
-,
UJ
X
t- U)
::::» ..., ~ 0
.5 ...
.
.... ,.
.... ..
01
I I
I I
I I
I I
I
0 10
20
30
40
50
60
70
AN
GLE
FROM
H
ORI
ZONT
AL
80
9
0
Fig
ure
5
.27
. P
lot
of
av
era
ge
ad
just
men
t fa
cto
rs
of
co
rnm
eal
as
a fu
ncti
on
o
f h
op
per
an
gle
(X)
(X)
3.0
I
2.5
a:
0 f- u
2.0
<
t lL
t- z 1
.5
L1J ~
1- en
::l
1.0
J 0 <
(
0.5
[
Q
l I
0 10
2
0
30
.A
NG
LE
0 2
.5
em
+
5.1
em
* 7
.6
em
tl 1
0.2
em
0
12
.7
em
0 1
5.2
em
.. -~
~ I
I---
__
L-
40
50
60
70
FROM
H
ORI
ZON
TAL
80
90
Fig
ure
5
.28
. P
lot
of
ad
just
men
t fa
cto
rs
of
co
rnm
eal
as
a fu
ncti
on
o
f h
op
per
an
gle
at
orif
ice
op
en
ing
o
f 2
.5 to
1
5.2
em
Table 5.5. Values of adjustment factor for cornmeal at various hopper angles
Orifice so· 60. 1o· so· Opening
(em)
2.5 0.271 0.273 0.290 0.296
5.1 0.248 0.258 0.254 0.307
7.6 0.284 0.256 0.270 0.294
10.2 0.280 0.267 0.291 -12.7 0.279 0.266 0.309 0.317
15.2 0.255 0.280 - 0.334
average 0.270 0.267 0.283 o. 310
89
26
'24
22
20 "" i.', '\. 18 ~
'../
~ 16 ~7. -v
r 14 ~ 0 0 i'. - J
~ 0 12 -Cf: L...'-/
10 ,, .... c '. 8 '-f ,
r -6
4
2
0 0
CORNf.lEAL
R2=.6706
2
0 0
0 0
0
0
0
0
0
4 6 8 10 (Tr.~ :.. sends)
MEASURED FLOI\ N,TE (g/s)
0
0 0
0
12 14
5.29. Plot of predicted versus measured mass flow rate of cornmeal
90
0
,...,. L:
r '-'
"" L.. ;--: < -~, : ._ -~ : c ~ - -!_ -
'-
, l
< 2
11
10 LINEAR LOW DENSITY POLYETHYLENE
9
8
7
6
5
4
3
2
:..---to=t---ID:r----£J
0 20 40 60 80
HOPPER ANGLE
0 2.5 em + 5.1 em o 7.6 em l:J. 10.2 em X 12.7 em ll 15.2 em
Figure 5.30. Plot of mass flow rates of LLDPE as a function of hopper angle at orifice opening of 2.5 to 15.2 em
91
The mass flow rate of LLDPE increases as the hopper angle and
orifice size increase. The rate of increase is higher for
large hopper angles and orifice sizes. The mass flow rate of
LLDPE is, therefore, a function of both hopper angle and
orifice size. In addition, the mass flow rate of LLDPE is
shown to be smaller than that of cornmeal and larger than that
of flour at a fixed orifice size.
The average adjustment factor for LLDPE as a function of
hopper angle, without considering the effect of orifice size,
is shown in Figure 5.31. Equations (4.4) and (4.5) accurately
estimate the values of the average adjustment factor for
LLDPE. The adjustment factor for LLDPE at each orifice size
as a function of the hopper angle is illustrated in Figure
5.32. The adjustment factor increases as the hopper angle
increases or orifice size decrease. From this figure, it is
clearly indicated that the adjustment factor is also a
function of orifice size. Table 5. 6 lists the values of
adjustment factor for LLDPE at various hopper angles. From
this table, it shows that the original model of Gregory and
Fedler (1987) almost entirely overestimates the flow rate of
LLDPE for hopper with two inclined side walls.
A plot of the predicted versus measured mass flow rate of
LLDPE is presented in Figure 5.33. It shows that the model
accurately predicts the flow rate of LLDPE and largely
eliminates the offset of the Gregory and Fedler (1987) model
92
\0
w
2.0~--------------------------------~
~ 1
.5
~
u <
LL
~
:z i
. 0 1
----
----
U
J ~
.__
en
::l
"""')
~ 0
.5
O'
I I
I I
I I
I J
I
0 10
20
30
40
50
60
70
~NGLE
FROM
H
ORI
ZON
TAL
80
90
Fig
ure
5
.31
. P
lot
of
av
era
ge
ad
just
men
t fa
cto
rs
of
LL
DPE
as
a fu
ncti
on
o
f h
op
per
an
gle
0----~--~--~--~--~--~----------~
0 10
20
3
0
40
5
0
60
70
AN
GLE
FROM
H
ORI
ZON
TAL
80
90
Fig
ure
5
.32
. P
lot
of
ad
just
men
t fa
cto
rs
of
LL
DPE
as
a fu
ncti
on
o
f h
op
per
an
gle
at
orif
ice
op
en
ing
o
f 2
.5 to
1
5.2
em
Table 5.6. Values of adjustment factor for linear low density polyethylene at various hopper angles
Orifice so· 60° 1o· 80° Opening
(em)
2.5 1.004 1.111 1.283 1.466
5.1 0.869 0.910 1.019 1.195
7.6 0.786 0.794 0.835 0.949
10.2 0.735 0.790 0.836 1.000
12.7 0.717 0.733 0.829 0.931
15.2 0.705 0.739 0.821 0.881
average 0.803 0.846 0.937 1.070
95
13
12
11
tO r-Ill '\. 9 Cl
v
I1J '3 \:;r-. ' Ill Ct:-o
'I ~ g [ J Vl _J J ~ 0 : [ ,-.L -t--1.!.!'-' \-u Cl lL! -Q: (L
3
)
0
Lit,=:_~ LOW DENSITY POLYETHYLENE
R2=.9765
0
2
0
0
4 6 (Thou~ : r: : s)
MEASURED FLOW f\ATE (g/s)
0
0
0
§
0
8 10
5.33. Plot of predicted versus measured mass flow rate of LLDPE
96
except under high flow rate condition. The raw data of mass
flow rate of LLDPE is presented in Table A.7.
Polymeric powders, such as PPS, tend to flow easily out
of the orifice with only a small amount of cohesion among
particles and adhesion of particles onto the hopper walls.
However, particle structure of PPS was not strong enough to
hold its shape when the flow took place. As a result,
particles were observed to undergo attrition upon impact,
resulting in the reduction of the particle size. Bulk
density, therefore, increased resulting in higher flow rates.
The mass flow rate of PPS at each orifice size as a
function of the hopper angle is illustrated in Figure 5.34.
The mass flow rate of PPS increases as the hopper angle and
orifice size increase. The rate of increase is higher for
large hopper angles and orifice sizes. The mass flow rate of
PPS is, therefore, a function of both hopper angle and orifice
size. In addition, the mass flow rate of PPS is shown to be
the largest among that of powdery materials investigated at a
fixed orifice size.
The average adjustment factor for PPS as a function of
hopper angle, without considering the effect of orifice size,
is shown in Figure 5.35. Equations (4.4) and (4.5) accurately
estimate the values of the average adjustment factor for PPS.
The adjustment factor for PPS at each orifice size as a
function of the hopper angle is illustrated in Figure 5.36.
The adjustment factor increases as the hopper angle increases
97
""' (/)
'\. (JI
v
""' w (/) r-u <{ ( n: 0
(/)
~ )
0 0 _J..(
u.r-v
Vl Vl (
~
0
14
13 POLYPHENYLENE SULFIDE
12
11
10
9
8
7
6
5
4
3
2
0 0 0 EJ
0 0 20 40 60 80
HOPPER Ak-L::
2.5 em + 5.1 em 0 7.6 em 6 . (\ -v ._ C~ X 12.7 em 'il 15.2 em
Figure 5.34. Plot of mass flow rates of PPS as a function of hopper angle at orifice opening of 2.5 to 15.2 em
98
1.0
1.0
4.0
-----------------------------------
3.5
~ 3
.0
._. u ~ 2
.5
t z 2
.0
UJ ~
~ 1
.5
:::>
J ~
1 . 0
~--------_.,.<j,---
0.5
0----~--~--~--~--~--~--~--~~
0 10
20
30
40
50
60
70
AN
GLE
FROM
H
ORI
ZON
TAL
80
90
Fig
ure
5
.35
. P
lot
of
av
era
ge
ad
just
men
t fa
cto
rs
of
PP
S
as
a fu
ncti
on
o
f h
op
per
an
gle
.......
0 0
4.0
I
0 2
.5
em
3.5
~
+
5.
1 em
~ 3
.0
* 7
.6
em
f-tl
1
0.2
em
u u:
2. 5
0
12
.7 e
m
0 1
5.2
em
1
- z 2
.0
UJ
:::E
~ 1
.5
:::>
J ~ 1
.0
0.5
O L
I I
j_·----~--~
0 10
20
Fig
ure
5
.36
.
30
40
50
60
70
80
90
AN
GLE
FROM
H
ORI
ZON
TAL
Plo
t o
f adj~stment
facto
rs
of
PP
S
as
a fu
ncti
on
o
f h
op
per
an
gle
at
ori
fice
op
en
ing
o
f 2
.5 to
1
5.2
em
or orifice size decrease. From this figure, it is clearly
indicated that the adjustment ~actor is also a function of
orifice size. Table 5. 7 lists the values of adjustment
factors for PPS at various hopper angles. From this table, it
shows that the original model of Gregory and Fedler (1987)
almost entirely underestimates the flow rate of PPS for hopper
with two inclined side walls.
A plot of the predicted versus measured mass flow rate of
PPS is presented in Figure 5.37. It shows that the model does
not accurately predict the flow rate of PPS and, therefore,
the large offset of the Gregory and Fedler (1987) model still
remains. The raw data of mass flow rate of PPS is presented
in Table A.a.
The coefficient A1 in equations ( 4. 5) and ( 4. 6) for
cornmeal, LLDPE and PPS as a function of the orifice size is
illustrated in Figures 5.38 to 5.40. These figures
illustrate that the coefficient A1 seems to be a superimposed
exponential and linear function for a wide range of orifice
sizes tested for all materials except for cornmeal. Moreover,
the coefficient A2 in equation (4.5) for cornmeal, LLDPE and
PPS as a function of the orifice size is illustrated in
Figures 5.41 to 5.43. The coefficient A2 appears to· be a
constant for each powder and flake material. Values for
cornmeal were much lower than those observed for other
materials. The results of coefficients A1 and A2 for LLDPE
101
Table 5.7. Values of adjustment factor for polyphenylene sulfide at various hopper angles
Orifice 40. so· 6o· 1o· so· Opening
(em)
2.5 1.083 1.154 1.299 1.422 -5.1 1.047 1.032 1.098 1.289 1.586
7.6 0.994 1.001 1.061 1.244 1.536
10.2 0.983 0.955 1.044 1.217 1.485
12.7 1.004 0.957 1.027 1.211 1.313
15.2 - 0.945 1.005 1.081 1.193
average 1.056 1.007 1.089 1.244 1.422
102
14
POLYPHENYLENE SULFIDE
12 R2=.8221
0 11
r.. 0 If) 10
" (Jl J 'J
w 9 1-r..
0 0 < If) 8 !rl) 0
~ g 0 Vl 7 0 _J ) u_ 0 0
0~ 6 Do W'J 0 1-0 5 0 0 0 w !r 4 0 0.. 0
0
3 0
B 2 0 0
~
0 0 2 4 6 8 10 12 14
(Thou3cnds) MEASURED FLOW ~TE (g/s)
5.37. Plot of predicted versus measured mass flow rate of PPS
103
4.0
3.5
3.0
2.5
.....
~ 2
.0
0 ~
I
1. 5
1.0
0.5
0
0 8
B
8 8
8
-.1
----J--.----'
0 2
4 6
8 10
1
2
14
16
O
RIF
ICE
O
PEN
ING
(e
m)
Fig
ure
5
.38
. P
lot
of
A1
co
eff
icie
nt
of
co
rnm
eal
as
a fu
ncti
on
o
f o
rif
ic
P o
pen
ing
siz
e
4.0
3.5
3.0
2.5
~
~ 2
.0
0 (J1
I 1
.5
1.0
0.5
0 0
~
... 0
ft
~-------L--------L--------L--------L------
--j
·------~
2
Fig
ure
5
.39
. 4 6
8 1
0
12
1
4
16
O
RIF
ICE
OPE
NING
(e
m)
Plo
t o
f A
1 co
eff
icie
nt
of
LL
DPE
as
a fu
ncti
on
o
f o
rif
ice
op
en
ing
siz
e
4.0
3.5
3.0
2.5
......
~ 2
.0
0 0'1
I
1. 5
1.0
0.5
0 0
o D
~
2 4
6 8
10
12
14
16
O
RIF
ICE
OPE
NIN
G
(em
) F
igu
re
5.4
0.
Plo
t o
f A
1 co
eff
icie
nt
of
PP
S
as
a fu
ncti
on
o
f o
rif
ice
op
en
ing
siz
e
4.0
3.5
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,_. ~ 2
.0
0 -...,J
I
1. 5
1.0
0.5
0 ~
o a ~
" --~----L----~--~--------
0 2
4 6
8 10
1
2
14
16
O
RIF
ICE
O
PEN
ING
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Fig
ure
5
.41
. P
lot
of
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eff
icie
nt
of
co
rnm
eal
as
a fu
ncti
on
o
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ice
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en
ing
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e
4.0
3.5
3.0
2.5
1-'
~ 2
.0
0 (p
1.5
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0.5
0
Q
g 0
0
0 •
~----~------_.------~------~------~------1------1----~
0 2
Fig
ure
5
.42
. 4 6
8 O
RIF
ICE
O
PEN
ING
10 (e
m)
12
14
16
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t o
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eff
icie
nt
of
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as
a f~nction
of
ori
fice
op
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ing
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e
.......
0 \0
4.0~----------------------------------
3.5
3.0
2.5
~ 2
.0
1. 5
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0.5
o
a a
o ,,
-0
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.
0 2
4 6
8 1
0
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14
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Fig
ure
5
. 4
3.
OR
IFIC
E
OPE
NIN
G
(em
) P
lot
of
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co
eff
icie
nt
of
PP
S
as
a fu
ncti
on
o
f o
rif
ice
op
en
ing
siz
e
and PPS as a function of orifice size are shown in Figures 4.1
and 4.2.
110
CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
The most significant result of this project is that it
appears that a universal correlation can be developed by
extending the Gregory and Fedler {1987) model to predict the
flow of granular materials (i.e., bulk solids and powders)
through rectangular orifices in a hopper device. The proposed
equations eliminate the need for recalibration for each
granular material. A second important result is the
illustration of the importance of the effect of hopper angle
on the flow rate of granular material through horizontal
orifices.
The flow of several types of granular material through
orifices has been investigated experimentally for a wide range
of operating conditions. The correlations of Gregory and
Fedler (1987) with minor modifications presented here, predict
the actual flow rate of several types of granular materials
through horizontal orifices in a hopper with two inclined side
walls with excellent agreement. These correlations have the
potential to be quite useful as a simplified approach to
describe the behavior of granular materials in common granular
storage facilities.
However, some difficulties were encountered during the
experiments with very high and low hopper angles (i.e. ,
vertical hopper angle and hopper angles smaller than the angle
111
of sliding). This was especially so for powdery materials
with very small particles size (i.e., less than 50 ~m) in
which large stagnant regions were observed and complete
stoppage of flow often occurred.
Because of the complex nature of granular flow phenomena,
it is recommended that the significance of other material
properties affecting the flow, which were not within the scope
of this project be studied. Some of these properties include
compressive strength, temperature, relative humidity, and
equilibrium moisture content which is especially important for
cohesive materials. Using a structured statistical analysis
technique, such as the Taguchi method, would prove to be very
valuable in assessing the relative importance of these
variables to granular flow. In addition, the behavior of
granular materials in the transition region between laminar
and turbulent flows should also be investigated.
112
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Zenz, F.A. 1976. Bulk Solids Efflux Capacity in Flooded and Streaming Gravity Flow. Fluidization Technology (D.L. Keairns et al., Eds.). Vol 2. pp. 239-252. Hemisphere Publishing. Washington, D.C.
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APPENDIX
RAW DATA OF MASS FLOW RATES
124
Table A.1. Mass flow rate* of polypropylene at various orifice sizes and hopper angles
Orifice 3o· 40° so· 6o· 70° Size (em)
2.5 1025.581 912.963 927.489 1094.823 1513.16
3.8 1824.531 1748.843 1827.323 2130.428 2725.00
5.1 2704.438 2725.004 2846.211 3168.329 3972.71
7.6 4451.403 4990.780 5594.440 6459.973 7443.84
8.9 5890.810 6171.320 6567.000 7224.450 8410.98
10.2 - 7841.956 8156.448 9071.847 10473.3
12.7 - 10965.10 11762.94 12919.95 14976.1
15.2 - - 15228.52 15796.75 17303.3
*Mean of five replications
125
Table A.2. Mass flow rate* of low density polyethylene at various orifice sizes and hopper angles
Orifice 40° so· 60° 1o· Size (em}
2.5 736.35 750.78 850.43 1132.25
5.1 2449.63 2476.33 2674.17 3197.64
7.6 4720.51 4745.20 5018.23 5851.47
10.2 6756.88 7017.84 7805.60 8864.00
13.3 10894.04 11303.7 12017.5 13103.8
*Mean of five replications
126
Table A.3. Mass flow rate* of reground nylon 6 at various orifice sizes and hopper angles
Orifice 30° 40° 50° 60° 70° Size (em)
2.5 552.11 524.45 540.28 686.57 -5.1 1997.23 1927.45 2093.50 2415.58 3159.70
7.6 4123.57 3986.65 4315.36 4883.17 6111.27
10.2 - 7673.56 8297.42 9112.35 10104.8
12.7 - 9071.85 9860.70 12370.70 -15.2 - 12006.86 12678.05 14273.89 -
*Mean of five replications
127
Table A.4. Mass flow rate* of dry sorghum at various orifice sizes and hopper angles
Orifice 3o· 40° so· 60° 70° Size (em)
2.5 - 1549.280 1786.889 2040.609 2500.96
5.1 - 3955.456 4431.598 5095.325 6022.61
7.6 7929.501 7906.770 7956.332 8443.116 10557.1
10.2 9928.589 10097.73 10592.36 12136.66 14331.5
12.7 12945.56 13842.02 14541.80 16563.77 17959.9
15.2 16968.92 17319.59 18670.96 20606.96 21339.7
*Mean of five replications
128
Table A.S. Mass flow rate* of flour at various orifice sizes and hopper angles
Orifice so· 60° 1o· so· Size (em)
8.9 1125.883 1954.125 2142.880 3369.86
10.2 1480.970 2577.443 3655.360 4425.25
12.7 2684.884 4824.768 6388.588 -15.2 6769.430 9152.980 - -
*Mean of five replications
129
Table A.6. Mass flow rate* of cornmeal at various orifice sizes and hopper angles
Orifice so· 6o· 70. so· Size (em)
2.5 574.058 578.780 613.670 625.425
5.1 1832.348 1907.073 1879.050 2276.25
7.6 4210.455 3788.965 3996.601 4354.03
10.2 6629.538 6329.505 6881.760 -12.7 9396.843 8938.478 10383.01 10668.7
15.2 11334.71 12442.68 - 14851.0
*Mean of five replications
130
Table A.7. Mass flow rate* of linear low density polyethylene at various orifice sizes and hopper angles
Orifice so· 60. 70. 80° Size (em)
2.5 549.809 608.287 702.191 802.667
5.1 1665.492 1743.578 1951.502 2289.58
7.6 3010.203 3041.889 3201.249 3635.22
10.2 4505.067 4845.695 5123.914 6127.27
12.7 6247.593 6383.893 7218.035 8103.93
15.2 8105.214 8496.871 9432.660 10122.9
*Mean of five replications
131
Table A.B. Mass flow rate* of polyphenylene sulfide at various orifice sizes and hopper angles
Orifice 40° so· 6o· 1o· so· Size (em)
2.5 576.973 614.637 692.011 757.477 -5.1 1952.315 1924.871 2047.494 2401.597 2956.00
7.6 3706.179 3731.888 3954.414 4637.953 5726.98
10.2 5861.787 5699.023 6229.785 7258.938 8855.44
12.7 8510.493 8108.281 8705.518 10262.89 11126.7
15.2 - 10575.76 11238.07 12090.58 13342.5
*Mean of five replications
132
