Chapter 1. Characterisation of solid particles
What is particle technology?
Techniques for processing and handling particulate solids. It plays a
major role in the production of materials in industry.
Chapter 1. Characterisation of solid particles
Individual solid particles are characterised by their size, shape, and
1.1 Particle shape
The shape of an individual particle is expressed in terms of the
sphericity F s, which is independent of particle size. The sphericity of a
particle is the ratio of the surface-volume ratio of a sphere with equal
volume as the particle and the surface-volume ratio of the particle. For a
spherical particle of diameter D p, F s =1; for a non-spherical particle,
the sphericity is defined as
Dp: equivalent diameter of particle
Sp: surface area of one particle
vp: volume of one particle
The equivalent diameter is sometimes defined as the diameter of a
sphere of equal volume. For fine particles, Dp is usually taken to be the
nominal size based on screen analysis or microscopic analysis. The
surface area is found from adsorption measurements or from the
pressure drop in a bed of particles. For many crushed materials, F s is
between 0.6 and 0.8. For particles rounded by abrasion, F smay be as
high as 0.95.
1.2 Particle size
In general "diameter" may be specified for any equidimensional
particles. Particles that are not equidimensional, i.e. that are longer in
one direction than in others, are often characterised by the second
longest major dimension. For needle like particles, Dp would refer to the
thickness of the particle, not their length. Units used for particle size
depend on the size of particles.
Coarse particles: inches or millimetres
Fine particles: screen size
Very fine particles: micrometers or nanometers
Ultra fine particles: surface area per unit mass, m2/g
1.3 Mixed particle sizes and size analysis
In a sample of uniform particles of diameter Dp, the total volume of the
particles is m/r p, where m = mass of the sample, r p = density. Since the
volume of one particle is vp, the total number of particle in the sample is
The total surface area of the particles is
To apply the above two equations to mixtures of particles having various
size and densities, the mixture is sorted into fractions, each of constant
density and approximately constant size.
1.4 Specific surface of mixture
If the particle density r p and spericity F s are known, the surface area of
particles in each fraction can be calculated and added to give the specific
where xi = mass fraction in a given increment,
= average diameter,
taken as arithmetic average of the smallest and largest particle diameters
1.5 Average particle size
(1). Volume-surface mean diameter,
, defined by
If the number of particles in each fraction Ni is known, then
(2). Arithmetic mean diameter
NT = number of particles in the entire sample
(3). Mass mean diameter
(4). Volume mean diameter
1.6 Number of particles in mixture
The volume of any particle is proportional to its "diameter" cubed.
a = volume shape factor
Assuming that a is independent of size
1.7 Screen analysis
Standard screens are used to measure the size (and size distribution) of
particles in the size range between about 3 and 0.0015in (76mm and
Screen is identified by meshes per inch, e.g. 10mesh, Dp = 1/10 = 0.1in.
The area of the openings in any one screen in the series is exactly twice
that of the openings in the next smaller screen. The ratio of the actual
mesh dimension of any screen to that of the next smaller screen is
Analysis using standard screen: Screens are arranged serially in a stack,
with the smallest mesh at the bottom and the largest at the top. Materials
are loaded at top and then shacked for a period of time (e.g. 20 minutes).
14/20: through 14 mesh and on 20 mesh
Screen analysis gives: xi and
Chapter 2. Motion of Particles through Fluids
2.1 Motion of particles through fluids
2.1.1 Mechanics of particle motion
Three forces acting on a particle moving through a fluid:
1). The external force, gravitational or centrifugal;
2). The buoyant force, which acts parallel with the external force but in
the opposite direction;
3). The drag force, which appears whenever there is relative motion
between the particle and the fluid
Drag: the force in the direction of flow exerted by the fluid on the solid is
2.1.2 Equations for one-dimensional motion of particle through fluid
Consider a particle of mass m moving through a fluid under the action of
an external force Fe. Let the velocity of the particle relative to the fluid
be u, let the buoyant force on the particle be Fb and let the drag be FD,
The external force can be expressed as a product of the mass and the
acceleration ae of the particle from this force,
The buoyant force is, be Archimedes’ law, the product of the mass of the
fluid displaced by the particle and the acceleration from the external
force. The volume of the particle is m/r p, the mass of fluid displaced is
(m/r p)r , where r is the density of the fluid. The buoyant force is then
Fb = mr ae/r p (3)
The drag force is
FD = CDu2r Ap/2 (4)
where CD is the drag coefficient, Ap is the projected area of the particle
in the plane perpendicular to the flow direction.
By substituting the forces into Eq(1), we have
Motion from gravitational force:
In this case, ae = g
Motion in a centrifugal field:
ae = rw 2
In this equation, u is the velocity of the particle relative to the fluid and is
directed outwardly along a radius.
2.2 Terminal velocity
In gravitational settling, g is constant. Also, the drag always increases
with velocity. The acceleration decreases with time and approaches zero.
The particle quickly reaches a constant velocity which is the maximum
attainable under the circumstances. This maximum settling velocity is
called terminal velocity.
In motion from a centrifugal force, the velocity depends on the radius
and the acceleration is not constant if the particle is in motion with
respect to the fluid. In many practical use of centrifugal force, du/dt is
small. If du/dt is neglected, then
Motion of spherical particles:
If the particles are spheres of diameter Dp, then
m = p Dp3r p/6
Ap = p Dp2/4
Substitution of m and Ap into the equation for ut gives the equation for
gravity settling of spheres:
2.3 Drag coefficient
Drag coefficient is a function of Reynolds number. The drag curve
applies only under restricted conditions:
i). The particle must be a solid sphere;
ii). The particle must be far from other particles and the vessel wall so
that the flow pattern around the particle is not distorted;
iii). It must be moving at its terminal velocity with respect to the fluid.
Particle Reynolds number:
u: velocity of approaching stream
Dp: diameter of the particle
r : density of fluid
m : viscosity of fluid
Stokes’ law applies for particle Reynolds number less than 1.0
CD = 24/NRe,p (13)
FD = 3p m ut Dp (14)
ut = g Dp2(r p - r )/(18m ) (15)
At NRe,p =1, CD =26.5 instead of 24 from the above equation.
Centrifugal: rw 2 ® g.
For 1000 < NRe,p <200,000, use Newton’s law
CD = 0.44 (16)
FD= 0.055p Dp2 ut2r (17)
Newton’s law applies to fairly large particles falling in gases or low
Terminal velocity can be found by trial and error after
guessing NRe,p to get an initial estimate of CD.
2.4 Criterion for settling regime
To identify the range in which the motion of the particle lies, the velocity
term is eliminated from the Reynolds number by substituting utfrom
If Stokes’ law is to apply, NRe,p <1.0. Let us introduce a convenient
Then NRe,p = K3/18. Setting NRe,p = 1 and solving for K gives K=2.6.
If K is less than 2.6 then Stokes’ law applies.
Substitution for ut using Newton’s law
NRe,p = 1.75K1.5
Setting NRe,p = 1000 and solving for K gives K = 68.9. Setting NRe,p =
200,000 and solving for K gives K = 2,360.
· Stokes’ law range: K < 2.6
· Newton’s law range: 68.9 < K < 2,360
· when K > 2,360 or 2.6 < K < 68.9, ut is found
using a value of CD found by trial from the
2.5 Hindered settling
In hindered settling, the velocity gradients around each particle are
affected by the presence of nearby particles. So the normal drag
correlations do not apply. Also, the particles in settling displace liquid,
which flows upward and make the particle velocity relative to the fluid
greater than the absolute settling velocity. For uniform suspension, the
settling velocity us can be estimated from the terminal velocity for an
isolated particle using the empirical equation of Maude and Whitmore
us = ut(e )n
Exponent n changes from about 4.6 in the Stokes’ law range to about 2.5
in the Newton’s law region. For very small particles, the calculated
ratio us/ut is 0.62 for e =0.9 and 0.095 for e =0.6. With large particles, the
corresponding ratios are us/ut = 0.77 and 0.28; the hindered settling
effect is not as profound because the boundary layer thickness is a
smaller fraction of the particle size.
If particles of a given size are falling through a suspension of much finer
solids, the terminal velocity of the larger particles should be calculated
using the density and viscosity of the fine suspension. The MaudeWhitmore equation may then be used to estimate the settling velocity
with e taken as the volume fraction of the fine suspension, not the total
Suspensions of very fine sand in water is used in separating coal from
heavy minerals and the density of the suspension is adjusted to a value
slightly greater than that of coal to make the coal particles rise to the
surface, while the mineral particles sink to the bottom.
Chapter 3. Size Reduction
Four commonly used methods for size reduction: 1). Compression; 2). Impact; 3).
Attrition; 4). Cutting.
3.1 Principle of size reduction
Criteria for size reduction
An ideal crusher would (1) have a large capacity; (2) require a small power input per
unit of product; and (3) yield a product of the single size distribution desired.
Energy and power requirements in size reduction
The cost of power is a major expense in crushing and grinding, so the factors that
control this cost are important.
3.2 Crushing efficiency
3.2.1 Empirical relationships: Rittinger’s and Kick’s law
The work required in crushing is proportional to the new surface created. This is
equivalent to the statement that the crushing efficiency is constant and, for a giving
machine and material, is independent of the sizes of feed and product. If the
sphericities a (before size reduction) and b (after size reduction) are equal and the
machine efficiency is constant, the Rittinger’s law can be written as
where P is the power required,
particle diameter before crushing,
and Kr is Rittinger’s coefficient.
is the feed rate to crusher,
is the average
is the average particle diameter after crushing,
Kick’s law: the work required for crushing a given mass of material is constant for the
same reduction ratio, that is the ratio of the initial particle size to the finial particle size
where Kk is Kick’s coefficient.
3.2.2 Bond crushing law and work index
The work required to form particles of size Dp from very large feed is proportional to
the square root of the surface-to-volume ratio of the product, sp/vp. Since s = 6/Dp, it
where Kb is a constant that depends on the type of machine and on the material being
The work index, wi, is defined as the gross energy required in KWH per ton of feed to
reduce a very large feed to such a size that 80% of the product passes a 100 m screen.
If Dp is in millimetres, P in KW, and
in tons per hour, then
If 80% of the feed passes a mesh size of Dpa millimetres and 80% of the product a
mesh of Dpb millimetres, it follows that
Example: What is the power required to crush 100 ton/h of limestone if 80% of the
feed pass a 2-in screen and 80% of the product a 1/8 in screen? The work index for
limestone is 12.74.
=100 ton/h, wi =12.74, Dpa =2 25.4=50.8 mm, Dpb =25.4/8=3.175 mm
3.3 Size reduction equipment
Size reduction equipment is divided into crushers, grinders, ultrafine grinders, and
cutting machines. Crusher do the heavy work of breaking large pieces of solid material
into small lumps. A primary crusher operates on run-of -mine material accepting
anything that comes from mine face and breaking it into 150 to 250 mm lumps.
A secondary crusher reduces these lumps into particles perhaps 6mm in
size. Grinders reduce crushed feed to powder. The product from an intermediate
grinder might pass a 40-mesh screen; most of the product from a fine grinder would
pass a 200-mesh screen with a 74 m opening. An ultrafine grinder accepts feed
particles no larger than 6mm and the product size is typically 1 to 5 m. Cutters give
particles of definite size and shape, 2 to 10mm in length.
The principal types of size-reduction machines are as follows:
A. Crushers (coarse and fine)
1. Jaw crushers
2. Gyratory crushers
3. Crushing rolls
B. Grinders (intermediate and fine)
Hammer mills; impactors
C. Ultrafine grinders
1. Hammer mills with internal classification
2. Fluid-energy mills
3. Agitated mills
D. Cutting machines
1. Knife cutters; dicers; slitters
Chapter 4. Mechanical Separations
Mechanical separations are performed based on the physical difference between
particles such as size, shape, or density. Mechanical separations are applicable to
heterogeneous mixtures, not to homogeneous solutions.
Screening is a method of separating particles according to size alone.
Undersize: fines, pass through the screen openings
Oversize: tails, do not pass
A single screen can make but a single separation into two fractions. These are called
unsized fractions, because although either the upper or lower limit of the particle sizes
they contain is known, the other limit is unknown. Material passed through a series of
screens of different sizes is separated into sized fractions, i.e. fractions in which both
the maximum and minimum particle sizes are known.
4.1.1 Screening equipment
Stationary screens and grizzlies; Gyrating screens; Vibrating screens; Centrifugal sitter.
Cutting diameter Dpc: marks the point of separation, usually Dpc is chosen to be the
mesh opening of the screen.
Actual screens do not give a perfect separation about the cutting diameter. The
undersize can contain certain amount of material coarser than Dpc, and the oversize can
contain certain amount of material that is smaller than Dpc.
4.1.2 Material balances over a screen
Let F, D, and B be the mass flow rates of feed, overflow, and underflow, respectively,
and xF, xD, and xB be the mass fractions of material A in the streams. The mass
fractions of material B in the feed, overflow, and underflow are 1- xF, 1- xD, and 1- xB.
FxF = DxD + BxB
Elimination of B from the above equations gives
Elimination of D gives
4.1.3 Screen effectiveness
A common measure of screen effectiveness is the ratio of oversize material A that is
actually in the overflow to the amount of A entering with the feed. These quantities
are DxD and FxF respectively. Thus
where EA is the screen effectiveness based on the oversize. Similarly, an
effectiveness EB based on the undersize materials is given by
A combined overall effectiveness can be defined as the product of the two individual
Example: A quartz mixture is screened through a 10-mesh screen. The cumulative
screen analysis of feed, overflow and underfolw are given in the table. Calculate the
mass ratios of the overflow and underflow to feed and the overall effectiveness of the
From the table, xF=0.47, xD=0.85, xB=0.195
4.1.4 Capacity and effectiveness of screens
The capacity of a screen is measured by the mass of material that can be fed per unit
time to a unit area of the screen. Capacity and effectiveness are opposing factors. To
obtain maximum effectiveness, the capacity must be small, and large capacity is
obtainable only at the expense of a reduction in effectiveness.
Filtration is the removal of solid particles from a fluid by passing the fluid through a
filtering medium, or septum, on which the solids are deposited. The fluid may be liquid
or gas, the valuable stream from the filter may be fluid, or the solid, or both.
Sometimes it is neither, as when waste solid must be separated from waste liquid prior
Filters are divided into three main groups: cake filters, clarifying filters, and crossflow
filters. Cake filters separate relatively large amount of solids as a cake of crystals or
sludge. Often they include provisions for washing the cake and for removing some of
the liquid from the solids before discharge. At the start of filtration in a cake filter,
some solid particles enter the pores of the medium and are immobilised, but soon
others begin to collect on the septum surface. After this brief period the cake of solids
does the filtration, not the septum; a visible cake of appreciable thickness builds up on
the surface and must be periodically removed. Clarifying filters remove small amount
of solids to produce a clean gas or a sparkling clear liquid such as beverage. The solid
particles are trapped inside the filter medium or on its external surfaces. Clarifying
filters differ from screens in that the pores of the filter medium are much larger in
diameter than the particles to be removed. In a crossflow filter, the feed suspension
flows under pressure at a fairly high velocity across the filter medium. A thin layer of
solids may form on the surface of the medium, but the high liquid velocity keeps the
layer from building up. The filter medium is a ceramic, metal, or polymer membrane
with pores small enough to exclude most of suspended particles. Some of the liquid
passes through the medium as clear filtrate, leaving a more concentrated suspension
4.3 The theory of filtration
In cake filters, the particles forming the cake are small and the flow through the bed is
slow. Streamline conditions are invariably obtained. From Kozeny equation,
where u is the velocity of the filtrate, L is the cake thickness, S is the specific surface of
the particles, is the porosity of cake, is the viscosity of the filtrate, and P is the
applied pressure difference. The filtrate velocity can also be written as
where V is the volume of filtrate which has passed in time t and A is the total crosssectional area of the filter cake.
For incompressible cakes can be taken as constant and the quantity 3/[5(1- )2S2] is
then a property of the particles forming the cake and should be constant for a given
Eq(3) is the basic filtration equation and r is termed the specific resistance. It is seen to
depend on and S. For incompressible cakes it is taken as constant, but it will depend
on the rate of deposition, nature of particles, and on forces between the particles.
In Eq(3), the variables V and L are connected, and the relation between them can be
obtained by making a material balance between the solids in the slurry and in the cake.
Mass in the filter cake is (1- )AL
is the density of the solids.
Mass of liquid retained in the filter cake is AL , where
is the density of the filtrate.
If J is the mass fractions of solids in the original suspension
If v is the volume of cake deposited by unit volume of filtrate then:
and from Eq(8):
Substituting for L in Eq(3)
Eq(11) can be regarded as the basic relation between P, V, and t. Two important
types of operation will be considered: 1). where the pressure difference is maintained
constant and, 2). where the rate of filtration is maintained constant.
Constant pressure difference
Eq(11) can be re-written as
Integrating Eq(12) gives
Thus for a constant pressure filtration, there is a linear relation between V2 and t.
Filtration at constant pressure is more frequently adopted in practical conditions.
Constant rate filtration
In this case,
P is directly proportional to V.
Flow of filtrate through the septum and cake combined
Suppose that the filter septum to be equivalent to a thickness Ls of cake, then if P is
the pressure drop across the cake and septum combined Eq(3) can be written as:
For constant rate filtration we have
For constant pressure filtration we have
4.4 Separations based on the motion of particles through fluids
Devices that separate particles of differing densities are known as sorting classifiers.
They use one of the two principal separation methods: sink-and-float and differential
4.4.1. Sink-and-float methods
A sink-and-float method uses a liquid sorting medium, the density of which is
intermediate between that of the light material and that of the heavy material. Then the
heavy particles settle through the medium, and the lighter ones float, and a separation
is thus obtained. This method has the advantage that, in principle, the separation
depends only on the difference in the densities of the two substances and is
independent of the particle size. This method is also known as the heavy-fluid
Heavy fluid processes are used to treat relatively coarse particles, usually greater than
10-mesh. A comment choice of medium is a pseudoliquid consisting of a suspension in
water of fine particles
4.4.2. Differential settling methods
Differential settling methods utilise the difference in terminal velocities that exist
between substances of different density. The density of the medium is less than that of
Consider particles of two materials A and B settling through a medium of density .
Let A be the heavier. If the smallest particle of A settles faster than the largest particle
of B, then complete separation of A and B can be achieved.
For settling in the Stokes’ law region, the terminal velocity can be calculated as
For equal-settling particles, utA = utB, therefore
For settling in the Newton’s law range
If the ratio of diameters of the smallest particle of A and the largest particle of B is
larger than the equal-settling ratio, then perfect separation of A and B can be achieved.