I have found some 'rules of thumb' describing when
jamming happens, i.e. when arcs form. I will include
some additional information, both to provide terminology
for eventual further study or questions and as search terms
for additional web searches.
This - Rheology of granular materials - is a vast topic which has
both scientific (physics-based) and engineering aspects (typical in
I will limit the references to the physics aspect of it, since they
use complex equations and simulations, but will mentions few.
The reason for this is that you mentioned 'explanation' as one of
The stress contours which are shown in some of these physics studies
eventualy determine shapes of the arcs. Pictures may give you some
for what is happening inside of the silo and also what is involved in
solving that kind of the equations (by FE method). You do not have to
make the computer runs yourself.
1) First a paper on a model system (very simplified system) which
to some very simple rules:
We study experimentally the jamming phenomenon of granular
flow of monodisperse disks of D = 5 mm diameter
in a two-dimensional hopper with opening R.
The jamming probability J(d) is measured where d[equivalent]R/D.
We found that J(d) decreases from 1 to zero when d increases from 2
to 5. From observing the disk configurations of the arch in the
jamming events, the jamming probability can be explained
quantitatively by treating the arch as the trajectory of a restricted
©2000 The American Physical Society
The tone of this article is a bit overdone.
The work is not that pioneering, other stidies were done,
nor unique but two things are of interest here:
a) a rule:
" No jams occurred when the opening was five times larger than the
disk diameter, but jams invariably occurred when the opening was twice
the disk diameter .."
b) the concept of the Jam probability J(d) .
I think that what you mean by 'Number of arches' -is 'arches which
happen under a given regime per month' or something like that. It was
not quite clear from the question 3 nor from your explanation.
It is ulikely that there will be some arc profile with multiple
minima, so I assume that you are after 'Jam probability' which is the
'average frequency of the arc formation' OK?
The paper itself, paper in Phys. Rev. Lett. 85, 5659
is available on-line but at a cost. That's not unusual in engineering
applications and that (business aspect) may explain why Jenike method
is not described on line. Jenike's company is giving courses on it,
as well solving the practical cases as consultants.(I can give the
link to their site, as well as references to books which explain the
method, an ASTM standard costs $35
and describe instruments used to characterise the materials).
Jenike's method calculates the J - the probability of arc formation
for a given hopper or silo and material It does not
calculates the shape of the arc. That can only be calculated by the FE
method - using computer - and we will not go into that.
2) This web article may be interesting (?)
It has a diagram on the stress conditions and references to more
3) This has experimental determination of the arc shape
for different materials - aand description of instrument.
(We do not consider irregual shape 'multi-arcs' (see above) right?
4) This is fairly technical paper
You may skip the equations, at look at Fig.9, 14, 15 which is giving
the stress countours and mobility of grains profiles. These will
shape of arcs. Also eq. 1 on page 56 may be quite useful. It
determines flow rate, which becomes zero when arc is formed. Rule is
that this happen when
D approaches k *d. The values of k and definitinxs of d and D are
Finaly, the rules of thumb I promised:
a) flow rate (and so arc formation) is independent of the H
provided H> 2.5 * W
(W= width of the cylinfer, D = aperture size H= head in the hopper)
and W> 2.5 * D and W > D+30*d
d is the grain size.
These rules are hidden on page 4405 in the intro of the above paper.
Please do remember that there are many other parameters, such as
conditioning,.. which modify the real situation, as described e.g.
conditioning of material
Rules of thumb are often good only for specific martials and
for which hey were developed.
5) This paper also is quite complex, but Figures provide some insights
6) Here is a paper which describes the material properties
These determine 'angle of repose' and flowability. There are diagrams
to determine the some such properties and flowability from that angle.
7) This model (described in reference 2, in Journal Nature) is
best mix of theory and simplicity for your application:
Simple model --Scalar Arching Model 
In a certain (quasi linear) limit, the model is completely soluble,
and has allowed us to obtain a remarkable fit of the experimental data
with only one free parameter .
I hope I have provided useful information and would appreciate some
feedback, either as rating or as request for clarification. In a case
like this, when there is a lot of info available, it is not always
to guess from a short question what is useful and what not.