Week # 11
MR Chapters 9 & 10
• Tutorial #11
• MR #9.1, 10.1.
• To be discussed on April 1,
2015.
• By either volunteer or
class list.
MARTIN RHODES (2008)
Introduction to Particle
Technology, 2nd Edition.
Publisher John Wiley & Son,
Chichester, West Sussex,
England.
Gas Cyclones
• During processing and handling of particulate solids,
separation of particles from suspension in a gas may be
required
• Generally, particles larger than about 100 mm can be
separated by gravity settling
• For particles less than 100 mm, more energy intensive
methods such as filtration, wet scrubbing and electrostatic
precipitation must be used
• Gas cyclones are best suited as primary separation devices
and for relatively coarse particles
• Electrostatic precipitator or fabric filter may be used
downstream to remove very fine particles
• Gas cyclones are generally not suitable for separation
involving suspensions with a large proportion of particles less
than 10 mm
• Most common type of cyclone
is known as reverse flow
cyclone separator
• Inlet gas is brought tangentially
into the cylindrical section
• A strong vortex is created
inside the cyclone body
• Particles in the gas are
subjected to centrifugal forces
which move them radially
outwards, against the inward
flow of gas and towards the
inside surface of the cyclone
• Direction of vortex flow
reverses near the bottom of
the cylindrical section
• Gas leaves the cyclone via the
outlet in the top
• Solids at the wall are pushed
downwards by the outer vortex
and out of the solids exit
• Gravity has been shown to
have little effect on the
operation of the cyclone
Flow Characteristics
• Rotational flow in the forced vortex within the cyclone body gives
rise to a radial pressure gradient
• This pressure gradient, combined with the frictional pressure losses
at the gas inlet and outlet and losses due to changes in flow
direction, make up the total pressure drop
• Pressure drop, measured between the inlet and gas outlet, is usually
proportional to the square of gas flow rate through the cyclone
• A resistance coefficient, the Euler number Eu, relates the cyclone
pressure drop Dp to a characteristic velocity:
• Where rf is the gas density
• The characteristic velocity v can be defined based on the crosssection of the cylindrical body of the cyclone
• Where q is the gas flow rate and D is the cyclone inside diameter
• The Euler number represents the ratio of pressure forces to the
inertial forces acting on a fluid element
• Value is practically constant for a given cyclone geometry,
independent of the cyclone body diameter
Efficiency of Separation
• Consider a cyclone to which solids mass flow rate is M,
mass flow discharged from the solids exit orifice is Mc
(known as the coarse product) and solids mass flow rate
leaving with the gas is Mf (known as the fine product)
• Total material balance on the solids may be written:
• Component material balance for each particle size x
(assuming no breakage or growth of particles within the
cyclone):
• Where dF/dx, dFf/dx and dFc/dx are the differential
frequency size distributions by mass (mass fraction of
size x) for the feed, fine product and coarse product
respectively
• Total efficiency of separation of particles from gas, ET, is
defined as the fraction of the total feed which appears in
the coarse product collected
• The efficiency with which the cyclone collects particles of
a certain size is described by the grade efficiency, G(x):
• Using the notation for size distribution described above:
• Combine to find an expression linking grade efficiency
with total efficiency of separation
• Above equation relates size distribution of feed, coarse
product and fine product
• In cumulative form, this becomes
• Consider a reverse flow
cyclone with a cylindrical
section of radius R
• Particles entering the
cyclone with the gas stream
are forced into circular
motion
• The net flow of gas is
radially inwards towards the
central gas outlet
• The forces acting on a
particle following a circular
path are drag, buoyancy
and centrifugal force
• The balance between these
forces determines the
equilibrium orbit adopted by
the particle
• Drag force is caused by the
inward flow of gas part the
particle and acts radially
inwards
• Consider a particle of diameter x and density rp following
an orbit of radius r in a gas of density rf and viscosity m
• Let the tangential velocity of the particle be U and the
radial inward velocity of the gas be Ur
• If we assume that Stokes’ law applies under these
conditions then the drag force is given by:
• The centrifugal and buoyancy forces acting on the
particle moving with a tangential velocity component U
at radius r are,
• Under the action of these forces, the particle moves
inwards or outwards until the forces are balanced and
the particle assumes its equilibrium orbit
• A relationship between U and the radius r for the vortex
in a cyclone is needed
• For a rotating solid body, U = rw, where w is the angular
velocity and for a free vortex Ur = constant
• For the confined vortex inside the cyclone body, it has
been found experimentally that the following holds
approximately:
• If we also assume uniform flow of gas towards the
central outlet,
• Combining,
• Where r is the radius of the equilibrium orbit for a particle
of diameter x
• If we assume that all particles with an equilibrium orbit radius greater
than or equal to the cyclone body radius will be collected,
• Then substituting r = R, we derive the expression below for the
critical particle diameter for separation, xcrit:
• Values of the radial and tangential velocity components at the
cyclone wall, UR and UR, may be found from a knowledge of
cyclone geometry and gas flow rate
• This analysis predicts an ideal grade efficiency curve
• All particles of diameter xcrit and greater are collected
• All particles of size less than xcrit are not collected
• In practice, gas velocity fluctuations and particle-particle
interactions result in some particles larger than xcrit being
lost and some particles smaller than xcrit being collected
• Consequently, the cyclone does not achieve such a
sharp cut-off as predicted by the theoretical analysis
• Grade efficiency curve for gas cyclones is usually Sshaped
• Particle size for which the grade efficiency is 50%, x50, is
often used as a single number measurement of the
efficiency of the cyclone
• x50 is also known as the equiprobable size since it is that
size of particle which as a 50% probability of appearing
in the coarse product
• In a large population, 50% of the particles of this size will
appear in the coarse product
• x50 is sometimes simply referred to as the cut size of the
cyclone
Scale-Up of Cyclones
• Scale-up of cyclones is based on a dimensionless group,
the Stokes number
• Characterizes the separation performance of a family of
geometrically similar cyclones
• Stokes number Stk50 is defined as:
• Where m is gas viscosity, rp is solids density, v is the
characteristic velocity and D is the diameter of the
cyclone body
• Physical significance of the Stokes number is that it is a
ratio of the centrifugal force (less buoyancy) to the drag
force, both acting on a particle of size x50
• For large industrial cyclones the Stokes number, like the
Euler number, is independent of Reynolds number
Range of Operation
• For a particular cyclone and inlet particle concentration,
total efficiency of separation and pressure drop vary with
gas flow rate as follows:
• Theory predicts that efficiency increases with increasing
gas flow rate
• In practice, total efficiency curve falls away at high flow
rates because re-entrainment of separated solids
increases with increased turbulence at high velocities
• Optimum operation is achieved somewhere between
points A and B, where maximum total separation
efficiency is achieved with reasonable pressure loss
• Position of point B changes only slightly for different
dusts
• Correctly designed and operated cyclones should
operate at pressure drops within a recommended range
• For most cyclone designs operated at ambient
conditions, this is between 500 to 1500 Pa
• Within this range, total separation efficiency ET increases
with applied pressure drop
• Above the top limit the total efficiency no longer
increases with increasing pressure drop and it may
actually decline due to re-entrainment of dust from the
dust outlet orifice
• It is therefore wasteful of energy to operate cyclones
above the limit
Storage and Flow of Powders
• In perfect mass flow, all the
powder in a silo is in motion
whenever any of it is draw from
the outlet
• The flowing channel coincides
with the walls of the silo
• Core flow occurs when the
powder flows towards the
outlet in a channel formed
within the powder itself
• Regions of powder lower down
in the hopper are stagnant until
the hopper is almost empty
• In mass flow, motion of powder is uniform and steady
state can be closely approximated
• Bulk density of the discharged powder is constant and
practically independent of silo height
• Stresses are generally low throughout the mass of
solids, giving low compaction of the powder
• No stagnant regions in the mass flow hopper
• Risk of product degradation is small compared with core
flow
• First-in-first-out flow pattern of mass flow hopper ensures
narrow range of residence times for solids in the silo
• Segregation of particles according to size is less of a
problem
• Friction between moving solids and hopper walls results
in erosion of the wall, which gives rise to contamination
of the solids by material of the hopper wall
• For conical hoppers, the slope angle required to ensure
mass flow depends on the powder-powder friction and
the powder-wall friction
• A hopper which gives mass flow with one powder may
give core flow with another
• In general, powders develop strength under the action of
compacting stresses
• The greater the compacting stress, the greater the
strength developed
• Gravity flow of a solid in a channel will take place
provided the strength developed by the solids under the
action of consolidating pressures is insufficient to
support an obstruction to flow
• An arch occurs when the strength developed by the
solids is greater than the stresses acting within the
surface of the arch
• The hopper flow factor, ff, relates the stress developed in
a particulate solid with the compacting stress acting in a
particular hopper
• A high value of ff means low flowability since high sC
means greater compaction
• A low value of sD means more chance of an arch forming
• Hopper flow factor depends on:
• Nature of the solid, nature of the wall material, slope of
the hopper wall
• Suppose that the yield stress (stress which causes flow) of
the powder in the exposed surface of the arch is sy
• This stress is known as the unconfined yield stress of the
powder
• If stresses developed in the powder forming the arch are
greater than the unconfined yield stress, flow will occur:
• This criterion may be rewritten as:
• The unconfined yield stress, sy, of the solids varies with
compacting stress, sC
• This relationship is called the powder flow function and is a
function only of the powder properties
•
The limiting condition for flow is:
•
This may be plotted on the same
axes as the powder flow function
to reveal the conditions under
which flow will occur for this
powder in the hopper
The limiting condition gives a
straight line of slope 1/ff
Where the powder has a yield
stress greater than sC/ff, no flow
occurs
Where the powder has a yield
stress less than sC/ff, flow occurs
For powder flow function (b), there
is a critical condition where
unconfined yield stress, sy, is
equal to stress developed in the
powder, sC/ff
This gives rise to a critical value of
stress, scrit, which is the critical
stress developed in the surface of
the arch
•
•
•
•
•
• The stress developed in the arch increases with the span
of the arch and the weight of solids in the arch
• Stress developed in the arch is related to the size of the
hopper outlet, B, and the bulk density, rB, of the material:
• Where H() is a factor determined by the slope of the
hopper wall and g is the acceleration due to gravity
• An approximate expression for H() for conical hoppers
is:
• The following are required for design for ensuring mass
flow from a conical hopper:
• (1) relationship between strength of powder in the arch,
sy (unconfined yield stress) with compacting stress
acting on the powder, sC
• (2) variation of hopper flow factor, ff, with: nature of the
powder (characterized by effective angle of internal
friction, d), nature of the hopper wall (characterized by
angle of wall friction, FW), slope of hopper wall
(characterized by semi-included angle of conical section,
or angle between sloping hopper wall and vertical)
• Hopper flow factor is therefore a function of powder
properties and hopper properties
• Knowing the hopper flow factor and powder flow
function, critical stress in the arch can be determined
and minimum size of outlet found corresponding to this
stress
• Mohr’s circle represents the
possible combinations of normal
and shear stresses acting on any
plane in a body under stress
• Mohr’s circle construction gives
the unconfined yield stress, sy
and compacting stress sC
• Experiments have demonstrated
that for an element of powder
flowing in a hopper:
• This property of bulk solids is
expressed by the relationship:
• Where d is the effective angle of
internal friction of the solid
• To examine the variation
of stress exerted on the
base of a bin with
increasing depth of
powder
• Assume that powder is
non-cohesive
• Consider a slice of
thickness DH at a depth H
below the surface of the
powder
• Downward force is
• Where D is the bin
diameter and sv is the
stress acting on the top
surface of the slice
• Assuming stress increases with depth, reaction of powder
below the slice acts upwards and is
• The net upward force on the slice is then
• If the stress exerted on the wall by the powder in the slice is
sh and the wall friction is tan FW, then the friction force
(upwards) on the slice is
• The gravitational force on the slice is
• Where rB is the bulk density of the powder, assumed to be
constant throughout the powder (independent of depth)
• If the slice is in equilibrium the upward and downward forces
are equal
• If we assume that horizontal stress is proportional to
vertical stress and does not vary with depth,
• As DH tends to zero,
• Integrating,
• If in general, the stress acting on the surface of the
powder is svo (at H = 0) the result is
• If there is no force acting on the free surface of the
powder, svo = 0
•
When H is very small
•
Equivalent to the static pressure at
a depth H in fluid density rB
When H is large,
•
•
•
•
•
And so vertical stress developed
becomes independent of depth of
powder above
Contrary to intuition, force exerted
by a bed of powder becomes
independent of depth if the bed is
deep enough
Hence most of the weight of the
powder is supported by the walls
of the bin
In practice, the stress becomes
independent of depth (and
independent of any load applied to
the powder surface) beyond a
depth of about 4D
• Rate of discharge of powder from an orifice at the base
of a bin is found to be independent of depth of powder
unless the bin is nearly empty
• Observation for a static powder that pressure exerted by
the powder is independent of depth for large depths is
also true for a dynamic system
• Confirms that fluid flow theory cannot be applied to the
flow of a powder
• For flow through an orifice in the flat-based cylinder,
experiment shows that:
• Where a is a correction factor dependent on particle size
• For cohesionless coarse particles free falling over a
distance h, their velocity neglecting drag and interaction,
will be
• If these particles are flowing at a bulk density rB through
a circular orifice of diameter B, then the theoretical mass
flow rate will be: