1. 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.

2. 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

3. 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

4. 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 cross-section 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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 S-shaped
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

12. 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

13. 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

14. 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

20. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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:

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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: