Papers in Physics, vol. 7, art. 070004 (2015)

Received: 17 November 2014, Accepted: 2 March 2015
Edited by: L. A. Pugnaloni
Reviewed by: G. Lumay, University of Liège, Belgium.
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.070004

www.papersinphysics.org

ISSN 1852-4249

Characterizing flowability of granular materials by onset of jamming in
orifice flows

Paul Mort1∗

This paper describes methods to measure flow rate and jamming onset of granules dis-
charged through a flat-bottom cylindrical hopper with a circular orifice. The intrinsic
jamming onset for ideal particles (spherical, monodisperse, smooth) is experimentally
measured by two independent methods, with good agreement. For non-ideal particles,
the normalized jamming onset increases with elongated granule shape, broadened size dis-
tribution and increased friction as measured by the drained angle of repose. An empirical
model of the jamming onset is introduced to quantify these effects over the range of mate-
rials investigated. The jamming onset can be used as a measure of differentiation between
relatively free-flowing granules.

I. Background

The motivation for this work is to find a means
of characterizing and differentiating flow quality of
relatively free-flowing granular materials. On the
one hand, industry requires methods that can be
performed relatively simply and reproducibly. On
the other hand, both industry and academia seek
better fundamental understanding of granular rhe-
ology via physical mechanisms governing flows; in
the case of industry, fundamental understanding
should extend to commercially-relevant granular
materials. This paper correlates the onset of jam-
ming with granular characteristics, including par-
ticle size distribution and particle shape. Details
are provided on size and shape characterization of
commercially-relevant materials.

Characterization of cohesive powder flow has
been relatively well established using shear cells

∗Email: mort.pr@pg.com

1 Procter & Gamble Co., 5280 Vine Street, Cincinnati, OH
45217, USA.

to quantify incipient bulk flow (i.e., yield loci) of
powders and granules. Flow functions calculated
using yield loci can differentiate between strongly-
cohesive, mildly-cohesive and free-flowing materi-
als [1]; this is highly relevant to bulk storage and
handling of powders. However, shear cell measure-
ments are relatively insensitive in regards to differ-
ences among freely-flowing granules. For applica-
tions where the quality of granular dynamic flow is
of interest, we need a more sensitive methodology,
hence the motivation of the current work. Various
approaches are discussed in the literature, includ-
ing using rotating drums [2, 3] and impeller-driven
flows [4, 5] to measure granular rheology. The cur-
rent work focuses on the onset of jamming in orifice
flows to differentiate the quality of otherwise freely-
flowing granules.

Let us consider jamming in the context of a shear
cell analysis. According to continuum modeling,
a “cohesionless” material measured using a shear
cell (i.e., a yield locus passing through the origin)
implies flow through a hopper cone opening of in-
finitesimal diameter. At this limit, the continuum
model fails because the orifice must be at least as

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Papers in Physics, vol. 7, art. 070004 (2015) / P. Mort

Figure 1: Schematic of a flat-bottom cylindrical
hopper device (FlodexTM).

large as the particle diameter. Further, local pack-
ing and jamming effects require the opening to be
significantly larger than a single particle [6]. A
theoretical analysis of intrinsic cohesion suggests
a boundary layer effect of 3-5 particle diameters
[7]; as a starting point, this can be taken as a the-
oretical minimum dimensionless orifice size for an
otherwise free-flowing granular material. Experi-
mental studies on the physics of jamming in gran-
ular flows report jamming transitions on the basis
of a dimensionless hopper opening at the onset of
the jamming transition [8, 9], where the dimension-
less hopper opening (Do/d) is defined by the ratio
of the opening size (Do) relative to a monodisperse
particle size (d). The current work is an empir-
ical investigation, seeking to elucidate the effects
of both size distribution and shape factors on the
jamming probability of free-flowing granules.

II. Experimental

The experimental device used in this work was a
flat-bottom cylindrical hopper with a set of inter-
changeable disks having a range of discrete orifice
sizes; it is available as a commercial flow testing
system, FlodexTM(Hanson Research, Chatsworth,
CA, USA), with additional disks machined as re-

Figure 2: Interpolation of Jamming Onset (J) at
25% mass discharged.

quired to extend the range of orifice diameters. The
range of orifice sizes used with the Flodex included:
2.0, 2.5, 3.0, 3.5, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18
and 20 mm. Experiments were done with common
materials including narrowly-classified glass beads,
Ottawa sand, and a variety of granular detergent
samples. In all cases, these samples were relatively
free-flowing according to shear cell measurements.

A schematic of the orifice flow instrument is
shown in Fig. 1. Approximately 100 ml of mate-
rial is used in the test, filling the 5.7 cm diameter
stainless-steel hopper to a height of about 4 cm by
pouring the granular sample through the loading
funnel. After the sample settles, the spring-loaded
discharge gate is opened and the sample is allowed
to drain through the orifice into a receiving cup be-
low. Once the flow stops and remains stopped for
30 seconds, the mass of discharged material is mea-
sured. Clogging is defined as a persistent jammed
state where the orifice remains obscured by the
granules at the point of flow stoppage. For each
measurement, the mass% discharged is calculated
according to the formula: (mass% discharged) =
100 × (mass discharged) / (sample mass). The av-
erage of the three mass% discharge calculations is
plotted as a function of the dimensionless orifice
size (Do/d), with the mass% discharged on the or-
dinate and the dimensionless orifice size on the ab-
scissa. This procedure is repeated using incremen-
tally larger orifice sizes until the hopper discharges
without clogging for three consecutive trials. The
averaged data are linearly interpolated to find the
Jamming Onset (J), defined here as the value of the

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Papers in Physics, vol. 7, art. 070004 (2015) / P. Mort

dimensionless orifice size at the point of 25 mass%
average discharge (Fig. 2).

For trials that drain without clogging, the
drained angle of repose, φd, is calculated accord-
ing to Eq. (1), derived assuming the granular ma-
terial discharges in the form of an inverted cone
with a base diameter equal to the hopper diameter,
Dh, and truncated where the cones apex protrudes
through the orifice with diameter, Do. Assum-
ing the remaining volume of retained material has
cylindrical symmetry (Fig. 3), simple solid geome-
try and the material’s repour (loose) bulk density,
ρbulk, are sufficient to relate the measured mass of
retained material, Mret, to the drained angle of re-
pose.

φd = arctan

[
24Mret

πρbulk(2D
3
h − 3DoD

2
h + D

3
o)

]
(1)

Orifice size is made dimensionless by scaling the
orifice size (Do) to a characteristic particle size (d).
The Sauter mean (d32) is used as the character-
istic size for this study. Note the Sauter mean is
weighted toward the surface area of the granules,
the surface area being relevant to inter-particle con-
tact and frictional interactions of the flow. Stud-
ies of cohesive dry powders show correlation be-
tween the Sauter mean size and flow properties [10].
Size distributions were measured by sieving, fitting
the size distribution using a log-normal distribu-
tion model to obtain a mass-based geometric mean
(d43) and geometric standard deviation (σg) for
each sample, then converting to the Sauter mean
using the Hatch-Choate relation [11], Eq. (2). Ex-
amples of size distribution characteristics are shown
in Fig. 4, for select samples A-D discussed in detail
later in the results section.

ln d32 = ln d43 − ln2 σg (2)

Shape characteristics were measured using an au-
tomated image analysis system, Solids Sizer (JM
Canty, Buffalo, NY), using statistical averaging of
104 cross-sectional particle images per sample. An
example of particles counted in a freeze-frame cap-
tured by the image analysis system is given in
Fig. 5. Note that the particles are in free-fall at the
point of image capture, so the cross sectional im-
age is randomly distributed over the possible states

Figure 3: Vertical cross section of residual mass
in a cylindrical hopper after draining successfully
without a clog.

Figure 4: Examples of particle size distribution
analysis for four samples, A-D. The data points are
from sieve analysis; the fitted curves are based on
a log-normal distribution, weighted to the mass on
each sieve.

Figure 5: Image analysis freeze frame and cross-
sectional aspect ratio (AR) analysis: a) grayscale
image; b) B&W image resulting from threshold
analysis with particles outlined in red, AR =
d1/d2.

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Papers in Physics, vol. 7, art. 070004 (2015) / P. Mort

Figure 6: Discharge rate data for Sample D, mea-
sured over a series of orifice diameters (4 mm to 9
mm). For each condition, the flow rate is the time
derivative of mass discharged. The flow rate data
are plotted according to the Beverloo relation in
Fig. 9.

of rotation. The median aspect ratio (AR50) was
used as the characteristic shape factor in this study.

The granular discharge rate was measured by
placing the receiver cup on an electronic balance
with data acquisition to record the discharge mass
with time. This experiment was done using orifice
sizes above the jamming onset. Discharge data are
provided in Fig. 6, using Sample D as an exam-
ple. Sample D is discussed in more detail in the
results section. The flow rate was determined by
taking the time derivative (slope) of the cumula-
tive discharge. The mass flow rate is converted to
a volumetric rate using the repour bulk density.

III. Results and Discussion

i. Analysis of jamming onset using a mul-
tivariate model

Results for nearly “ideal” materials including
tightly classified glass beads and washed Ottawa
sand are shown on Fig. 7. These samples are nearly
mono-disperse. The glass beads are nearly spher-
ical, while the sand is slightly irregular in shape.
Both have relatively smooth-surfaces and a low
drained friction angle. The jamming onset for the
glass beads is in the range of 3 to 5 particle di-
ameters. The washed sand has a slightly higher
jamming onset of about 7 diameters. Other sam-

Figure 7: Cylindrical hopper discharge data, jam-
ming onset calculations and drained angle of repose
for glass bead and washed sand samples.

ples characterized in this work included commercial
granules having similar compositions, but with a
range of particle size and shape characteristics, the
effect of which resulted in relative jamming onsets
up to 30 and drained friction angles in excess of
60 Degrees. Because the commercial samples have
several aspects of variability, a multivariate model
was formulated to assess the relative effects of fric-
tion, size distribution and shape characteristics.

Data from the broader set of 23 samples (Ta-
ble 1), including commercial granular materials
and manipulations thereof, were used to gener-
ate a multivariate power-law model for the jam-
ming onset (J) as a function of particle parame-
ters [Eq. (3)]. The model parameters include the
shape factor (median aspect ratio, AR50), size dis-
tribution breadth (geometric standard deviation,
σg) and an excess friction factor defined as XF =
max[1, tan(φd)]. The range in breadth of size dis-
tribution was from about 1.0 to 2.2; the range of
median aspect ratio was from about 1.0 to 1.4. The
fitted coefficients (a, b, c) represent exponents in
the power-law form of the model. While this equa-
tion is purely empirical, its form is logical in the
sense that it reduces to an intrinsic jamming inter-
cept (kj) when the other terms are minimized. In
other words, extrapolation to kj represents the jam-
ming onset of an idealized sample (mono-disperse,
spherical, smooth, low friction).

ln(J) = ln(kj) + a ln(AR50)

+ b ln(σg) + c ln(XF) (3)

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Figure 8: Multivariate regression of jamming onset
model [Eq. (3)]; deviation from the diagonal rep-
resents uncertainty in the empirical model. Solid
data points labeled A-D indicate samples used in
Flow Rate analysis (Fig. 9); their size distributions
are shown in Fig. 4.

Figure 9: Regression [Eq. (5)] of flow rate data
collected for selected samples (A-D) having a range
of jamming onsets.

The results of the multivariate model are shown
graphically in Fig. 8. While the correlation coeffi-
cient (∼0.86) indicates that the model is reason-
ably predictive, the scatter suggests uncertainty,
perhaps in the choice of model parameters, their
interaction and/or uncertainty in measurements.
For example, the model may be over-simplifying
the parameter space, ignoring potentially impor-
tant factors such as more complex shape factors,
the breadth of the shape distribution, and interac-
tions between size and shape distributions.

The statistical analysis is shown in Table 2. The
most statistically significant factors are the intrinsic
jamming intercept, kj, excess friction factor, XF ,
and the geometric size distribution, σg. The value
of kj is ∼2.9, at the lower end of the range predicted
by Wier [7]. While the particle shape factor, AR50,
is somewhat less significant statistically, its high
coefficient (2.38) indicates that particle shape can
have a strong impact on jamming, even over the
relatively narrow range tested.

ii. Flow rate measurements and Beverloo
analysis

Flow rate data were analyzed using the Beverloo
equation [12]. It is shown here in volumetric form
[Eq. (4)], where V ′ is the volumetric feed rate, Do is
the orifice diameter, d is the characteristic particle
size (Sauter Mean), g is the gravitational constant
and C and kf are fit parameters; Do − kfd rep-
resents a reduced orifice size for active flow caused
by a boundary layer that scales with doparticle size.
For the purpose of regression analysis as a function
of relative orifice size (Do/d), the Beverloo equation
can be written in linearized form, Eq. (5), where
C is solved using the regression slope and kf using
the intercept.

V ′ = C
√
g (Do −kfd)

5/2
(4)

V ′2/5

d
= C2/5g1/5

(
Do
d
−kf

)
(5)

Several flow rate data sets are shown in Fig. 9.
These data sets (Samples A-D) represent similar
materials, but with different shape and size distri-
bution characteristics. Sample A is a commercial
granule; samples B and C are manipulated by clas-
sification; and sample D is classified then further

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Table 1: Experimental data for relative jamming
onset (J) as a function of excess friction factor
(XF), geometric size distribution (σg), and median
aspect ratio (AR50).

Sample J XF σg AR50
A 18.6 1.49 1.84 1.36
B 16.2 1.01 1.76 1.35
C 11.8 1.05 1.63 1.35
D 7.9 1.00 1.37 1.26
E 6.8 1.00 1.33 1.30
F 7.1 1.00 1.34 1.24
G 8.9 1.10 1.64 1.25
H 9.6 1.08 1.48 1.32
I 28.9 1.90 1.79 1.41
J 15.1 1.00 2.14 1.26
K 18.3 1.53 1.50 1.32
L 28.2 1.24 1.79 1.34
M 15.6 1.00 1.56 1.41
N 19.1 1.83 1.56 1.41
O 6.5 1.08 1.47 1.28
P 7.3 1.00 1.39 1.27
Q 9.1 1.05 1.41 1.26
R 7.6 1.09 1.32 1.27
S 17.6 1.73 1.31 1.26
T 6.8 1.00 1.23 1.21
U 7.1 1.00 1.10 1.20
V 4.3 1.00 1.05 1.10
W 3.5 1.00 1.05 1.15

Table 2: Parameter estimates and standard errors
obtained from multi-variate regression of Eq. (3)
using the data of Table 1. “Prob> |t|” is the proba-
bility that the true parameter value is zero, against
the two-sided alternative that it is not; values less
than about 0.05 are typically regarded as highly
significant.

Term Estimate Std Error P> |t|
ln(kj) 1.072 0.225 0.0001
a 2.380 1.267 0.0758
b 1.404 0.398 0.0023
c 1.095 0.265 0.0006

rounded in a layering process. The jamming onsets
(J) of these samples span a significant range from
about 8 to 20. However, the Beverloo regressions
are remarkably similar. All lie on a common slope
(C ∼ 0.57) and extrapolate to a similar zero-flow
intercept (kf ∼ 2.64). In other words, the granular
flow behavior is consistent for orifice sizes above the
jamming onset, but subtle differences in the gran-
ular characteristics and frictional properties have
a significant effect on the onset of jamming. This
suggests that the onset of jamming is a useful mea-
sure to differentiate the flow behavior of relatively
free-flowing granular materials.

Consider kf to represent an “intrinsic jamming
onset” at which the flow rate goes to zero. Note
that the extrapolated value for intrinsic jamming
(i.e., no flow) is only slightly lower than the power-
law intercept of the multivariate model (kj ∼ 2.9).
Recall, the measurement of the jamming onset, J,
is based on the interpolated orifice size for an av-
erage 25% sample discharge from the test hopper.
Thus, it is logical that kj, which is based on multi-
variate analysis of J, may slightly exceed kf , which
implies zero flow. On the other hand, the slope
of the jamming probability function tends to be
quite steep in the range from zero to 25% discharge.
So, for the purpose of this paper, we can merge
the interpretation of kj and kf —both represent
an intrinsic jamming limit. Indeed, even though
each one was measured independently using differ-
ent methods, their values are nearly identical.

An interpretation of this result is that the “in-
trinsic jamming onset” is characteristic of orifice
flow for an ideal particle (monodisperse, spherical,
no excess friction), and that differences between the
intrinsic onset and the actual measured jamming
onset are due, in large part, to irregularities in the
size distribution, shape and frictional roughness of
the granules.

IV. Conclusions

The statistical analysis of jamming is a promis-
ing approach for characterization of relatively free-
flowing granular materials. In this work, we eval-
uated the effect of dimensionless orifice size on the
onset of jamming and flow rate. A multivariate
analysis of the jamming onset suggests an intrinsic
jamming probability of about 2.9 particle diame-

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Papers in Physics, vol. 7, art. 070004 (2015) / P. Mort

ters for an idealized sample; irregularities such as
size distribution, shape or excess friction support
jamming across an increased number of particle
diameters. While all three parameters are signif-
icant in the regression analysis, the result suggests
that shape (aspect ratio) is an especially impor-
tant contributor to jamming. Analysis of the flow
rate data using the Beverloo equation extrapolates
to a similar value for the intrinsic jamming onset
(∼2.6 particle diameters), providing good experi-
mental agreement between two independent meth-
ods.

The experimental methods described herein are
relatively simple. Continued work covering a wider
range of materials can help to build stronger empir-
ical models for the effects of shape, size distribution
and other frictional properties on jamming proba-
bility. In addition, first principle modeling of dense
flows may help to elucidate the underlying mecha-
nisms of jamming and the real effects of non-ideal
particle characteristics, and perhaps help to build
a more theoretical model.

Lastly, the reader should temper the results pre-
sented with the caveat that all of the samples used
herein are relatively free-flowing granular materials.
As materials become more cohesive and/or com-
pressible, it is not clear that the flow rate and jam-
ming onset experiments will continue to converge
on a common intrinsic jamming onset.

Acknowledgements - The original data for this
paper were collected and analyzed as part of a
student internship at Procter & Gamble by Derek
Geiger, University of Michigan, with the assistance
of Mark Wandstrat at P&G, and presented at the
2007 Annual Meeting of the American Institute of
Chemical Engineers [13]. This work at P&G fol-
lowed an earlier collaboration with Professor R. P.
Behringer [14]; the author acknowledges Professor
Behringer’s guidance and insight on using funda-
mental granular physics to address problems of in-
dustrial relevance.

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