Principles of Powder Mechanics. Essays on the Packing and Flow of Powders and Bulk Solids

232 Pages·1970·4.308 MB·English

by  R. L. Brown, J. C. Richards and P. V. Danckwerts (Auth.)

PRINCIPLES OF
POWDER MECHANICS
Essays on the Packing and Flow
of Powders and Bulk Solids
R. L. BROWN
AND
J. C. RICHARDS
PERGAMON PRESS
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First edition 1970
Library of Congress Catalog Card No. 73-99993
Printed In Hungary
08 006 605 4
PREFACE
THIS monograph is based on four postgraduate lectures delivered by
one of us (R.L.B.) at the Imperial College of Science and Technology,
London, in May 1962. Much of the stimulus to present an orderly
treatment of the subject is due, therefore, to Dr. J. H. Burgoyne who
arranged for the lectures to be given.
Since that date a report of a working party of the Institution of
Chemical Engineers, The Storage and Recovery of Paniculate Solids,
edited by the other of us (J.C.R.), has been published. This report
surveyed the field with particular emphasis on practical matters, leaving
the reader to thread his own way through the details that he could
seek among the references. This monograph sets out as connected a
view as possible of what we believe to be the fundamentals of the
subject.
Powder mechanics is an inter-disciplinary subject. Many practical
problems have been solved by engineers in the mineral processing and
chemical industries. An art of erecting storage vessels has been devel-
oped by structural and civil engineers. The factors governing cohesive
and frictional forces have been studied by physical chemists and physi-
cists. Some of the mathematics has been provided by students of soil
mechanics, aided latterly by the computer. Early work, however, cannot
be disregarded. A complete account of this varied literature, covering
adequately its engineering, physical, and mathematical content, does
not seem to be warranted at the present time, when the subject is advanc-
ing rapidly. It has, however, seemed worth while to select from among
ix
X PREFACE
the literature those findings that lead to an understanding of the physical
principles on which a powder mechanics may be based.
Some hitherto unpublished results have been included by permission
of the British Coal Utilisation Research Association.
The BCURA R. L. BROWN
Randalls Road J. C. RICHARDS
Leatherhead
Surrey, England
ACKNOWLEDGEMENTS
THE permission of the following to reproduce material is gratefully
acknowledged: The American Society of Agricultural Engineers (Agri-
cultural Engineering). The American Ceramic Society. The American
Society of Mechanical Engineers. The British Coal Utilization Research
Association. Butterworths (Fuel). The Cement and Concrete Associa-
tion (Magazine of Concrete Research). Dr. Dietrich Steinkopff Verlag,
Darmstadt (Rheologica Acta). Engineering. The Faraday Society. The
Institution of Chemical Engineers. The Institution of Civil Engineers
(Geotechnique). The Pharmaceutical Society of Great Britain (Journal
of Pharmacy and Pharmacology). Macmillan (Journals) Ltd. (Nature,
London). McGraw-Hill Publications (ChemicalEngineering) .The North
Holland Publishing Co. The Safety in Mines Research Establishment,
Sheffield (and the Ministry of Power). The Society of Chemical Industry.
Technische Hogeschool, Delft. University of Minnesota, the Institute
of Technology. University of Utah, the Engineering Experimental
Station.
Special thanks are also given to the British Coal Utilization Research
Association for the provision of facilities during the preparation of the
manuscript and to Mrs. K. Fortescue who typed many of the early
drafts.
xi
CHAPTER 1
INTRODUCTION
AN UNDERSTANDING of the factors governing the packing and flow of
powders and bulk solids is needed in mineral processing and in many
chemical industries. Storage in bins, the formation of loosely coherent
aggregates, and the manufacture of dense strong compacts require in-
formation on the geometry of packings and the transmission of forces
through them. Discharge through bin outlets, flow through hoppers
and chutes, flow in mixers, and the filling of die cavities require infor-
mation on flow patterns, strength of powders, and their adherence to
surfaces. In these processes, the particles and granules are substantially
in contact with each other, and what happens within the assembly is
largely governed by inter-particle cohesion and friction. In other pro-
cesses involving solids, a fluid phase is dominant, e.g. flow through
fixed beds, fluidized systems, some systems of hydraulic and pneumatic
transport, and dust catchers: in this case design and operation of the
equipment is aided by knowledge of fluid mechanics. Likewise, know-
ledge of a powder mechanics is needed when the solids phase is domi-
nant.
The elements of powder mechanics have long been known. Recalling
that Hagen reported on the flow of sand in 1852 and Osborne Reynolds
observed dilatancy during deformation of a mass of sand in 1885, the
subject might well have developed side by side with fluid mechanics.
This did not happen. Only now, when new powders are being developed,
process equipment is being automated, and fine powders are handled
in large quantities, has the need become pressing.
The fundamental equations of powder mechanics are the same as
those of soil mechanics. They are based on the early work of Coulomb
1
2 PRINCIPLES OF POWDER MECHANICS
(1776) and Rankine (1857) on the frictional behaviour of a mass of
sand. The differences between the two subjects are important. Since
some powders, known from their behaviour to be cohesive, would be
regarded in soil mechanics as free flowing, test methods suitable for
use with powders are needed. Useful advances in this direction have
been made by Dawes (1952) and Jenike et al. (1960). In most cases the
boundary conditions in powder mechanics are not the same as in soil
mechanics. Jenike (1961) and Richmond and Gardner (1962) have
worked out some appropriate solutions. The third difference is that
powders can be subjected to much larger deformations than is common
in soil mechanics. Geniev (1958) has investigated equations of motion,
and Brown (1961) has proposed an energy theorem for freely flowing
granules.
During the past 50 years many studies have been made of the flow
patterns of powders in bins and hoppers and of rates of discharge
through apertures. It has been rare for these studies to include measure-
ments of the pressure distribution within the powder and at the walls
of the vessel. This serious defect is now being remedied. It has meant
that powder mechanics has not been developed far enough to enable
the engineer to design and operate his equipment in the way he is
accustomed with fluid systems. For this reason, no attempt is made
here to cover the civil engineering design and erection of structures for
storing bulk solids and powders: for information on this art, reference
should be made to the appropriate authorities, e.g. Ketchum (1919) and
Reimbert and Reimbert (1961). Powder mechanics is developing rapidly
at the present time, and it is to be expected that theoretical and experi-
mental connections will soon be established between flow patterns
and stress distributions. The physicist has an important contribution
to make here. Meanwhile a considerable literature is available, and it is
useful to see what general principles have emerged.
THREE PRINCIPLES
In a paper published in 1885, Reynolds observed that a "tightly
packed mass of granules enclosed within a flexible envelope invariably
increases in volume when the envelope is deformed: if the envelope
is inextensible but not inflexible, no deformation is possible until the
INTRODUCTION 3
applied forces rupture the bag or fracture the granules". This quotation
expresses a fundamental principle of dilatancy. A simple consequence
(Jenkin, 1931) of this geometrical property of a powder or bulk solid
is that shearing causes some granules, previously in contact, to separate
so that the contacts become "slack": surfaces of sliding, permitting
relative displacements of granules, are thereby formed.
The second principle may be called the principle of mobilization of
friction. Ignoring its direction, the frictional force between any two
granules in a powder at rest can take any value between zero and a
limiting value, the latter being reached when the granules are just about
to move relative to each other whilst remaining in contact: the limiting
value depends on the normal force between the granules. It follows that
the stress distribution in a powder at rest is indeterminate. Another
simple consequence is the occurrence of a range of equilibrium states,
and hence of bulk densities and angles of repose. If the frictional force
due to shear of a powder reaches its limiting value, a surface of sliding
is formed.
The microstructure of a surface of sliding is not known for certain.
The principle of dilatancy permits a surface of sliding to occur when
there are slack contacts between the particles, and thus the tangential
and normal stresses are zero. In contrast, the principle of mobilization
of friction leads to a macroscopic picture in which there can be a dis-
continuity in the stresses or in one of their derivatives with respect
to displacement. These two views are compatible, however, if the occur-
rence of slack contacts is a transient phenomenon; their role may be
connected with the initiation of surfaces of sliding.
As a powder accelerates from rest to a flowing condition, the granules
are rearranged progressively. Since flow through apertures is steady,
it seems that the state of the powder in motion can be determinate.
When this is the case, there must be a restriction on the generality of
the principle of mobilization of friction in that the energy is minimized
in a well-defined manner. This is the third principle of minimum energy
of flowing granules. A consequence is that discharge rates through
apertures can be calculated from the shape of the surfaces of sliding at
the aperture.
4 PRINCIPLES OF POWDER MECHANICS
SOME DEFINITIONS
Bulk solids composed of a wide variety of materials are difficult to
define closely. This is not only because small variations in some of
the primary properties can result in very different behaviour, but also
because secondary properties not directly associated with the particles
themselves can have overriding importance. Thus the presence or ab-
sence of moisture, the severity of prior compaction, the ambient tem-
perature, or the proximity of continuous vibrations can, among other
conditions, be more important than the size, shape, hardness, particle
density, or surface roughness of the components of the powder. In the
present state of knowledge it is sufficient to have available a very gen-
eral—almost intuitive—set of definitions. The terms proposed below
are sufficient for a discussion of the experimental evidence available.
The distinctions between bulk solids are based largely on Chapter 2 of
The Storage and Recovery of Paniculate Solids (Richards, 1966).
A bulk solid is an assembly of discrete solid components dispersed in a
fluid such that the constituents are substantially in contact with near
neighbours. This definition excludes suspensions, fluidized beds, and
materials embedded in a solid matrix. It is, however, very general
and, depending upon the size of the constituents, may be considered
in the following subdivisions:
A powder is composed of particles up to 100 μ in size with the further
division that ultra fine, super fine, and granular powders contain
ultra-fine (0-1-1-0 μ), super-fine (1-10 μ), or granular (10-100 μ)
particles. The term fine powder is used to imply the presence of a
wider range covering super-fine and granular particles.
A granular solid is composed of granules ranging from about 100 to
3000 μ.
A granular material covers the combined range of granular powders
and granular solids. Thus with components ranging in size from
about 10 μ up to 3 mm, this term covers most of the materials
used in laboratory experiments.
A broken solid contains grains and lumps almost all of which are
larger than 3 mm.
INTRODUCTION 5
Cohesion is the sticking of the components of a bulk solid to one another
and is conveniently assessed as the resistance of a powder to shear
at zero compressive normal load.
Adhesion is the sticking of a bulk solid to a wall or substrate.
Tensile strength is the force, per unit area of broken face, required to
split a bulk solid compact at zero shear in the plane of the broken
face.
Angle of repose is the angle to the horizontal assumed by the free surface
of a heap at rest and obtained under stated conditions.
The poured angle of repose is formed by pouring the bulk solid to
form a heap below the pour point.
The drained angle of repose is formed by allowing a heap to emerge as
superincumbent powder is allowed to drain away past the peri-
phery of a horizontal flat platform previously buried in the powder.
Angle of sliding is the angle of elevation of the surface separating sub-
stantially stationary material from flowing material measured close
to the escape aperture through which the bulk solid is flowing. The
shape of the containing vessel and aperture must be stated.
Fractional solids content is the ratio of the apparent powder density to
the effective particle density, or
bulk density
solids density
Voidage is unity minus the fractional solids content.
In the course of examining the properties of bulk solids it will be
necessary to introduce the yield locus, coefficient of internal friction,
wall-yield locus, effective locus, and other terms of special application.
It is convenient to delay their definitions until they can be discussed at
length.
Particle size has not been defined because it cannot be described
adequately in a sentence. There are many ways of assessing the "size" of
a particle, and only two of these are used to any extent here. The sieve
size is the most convenient to obtain, and is adequate for particles not
much smaller than about 50 μ (or 5 μ if air-jet sieves are available).
BR-PPM 2