SOLIDS NOTES 11, George G. Chase, The University of Akron
11. GRADE EFFICIENCY
Classification is the art of separating solid particles in a mixture of solids and fluid into fractions according
to particle size or density by methods other than screening. Most methods of separation are not 100%
effective. There is usually a range of particle sizes that are separated with varying degrees of efficiency.
The variation in the efficiency is referred to as the Grade Efficiency Curve. A good discussion on Grade
Efficiency is given by Svarovsky.1
The grade efficiency is a way of characterizing how well particles are separated according to size (density,
or some other desired property). Separation of particles by size is referred to as classification.
The process of classification has been used for many years. Some methods make a sloppy separation while
others make a sharp classification. Usually for the same production rate, the sharp separation will be more
expensive, as could be deduced from the 2nd Law of Thermodynamics.
11.1 Why Do We Classify Particles?
We classify particles to remove contaminants and unlike particles (wheat from chaff, or metal particles
from polymers). We classify to remove unwanted parts of a size distribution. In Figure 11-1 indicates the
tailings and oversize particles that may be removed from a material for a particular purpose.
FREQUENCY
SIZE
TAILINGS
OVERSIZE
Figure 11-1. Size distribution indicating undesired tailings and oversize particle ranges.
1
Ladislav Svarovsky, Solid-Liquid Separation, 3rd ed., Butterworths, London, 1990, Chapter 3.
11-1
SOLIDS NOTES 11, George G. Chase, The University of Akron
There are many processes that may be used to classify particles:
Screening
These methods are effective, but rate of production
may be low.
Sieving
Air or water classification
These methods take advantage of different drag
forces acting on different size particles.
Physical separations
These methods rely on physical properties other
than differences in fluid drag to classify.
•
•
•
•
•
Magnetic
Gravity
Electrostatic
Radiation
Color
An example of gravity separation if a mixture of plastic milk jugs (high density poly ethylene, density of
0.95 g/cc) and plastic ware (polystyrene, density of 1.05 g/cc) is dumped into a tank of water, the milk jug
plastic will float while the plastic ware will sink. This provides a simple means for separation. Some
additional examples of separators are given in Figures 11-2 through 11-4.
Paper
Exit
Stream
Air
Solid Trash
Feed
Stream
Air
Air
Plastic and
metals input
Metals and
Plastics Exit
Stream
Figure 11-2. Air classifier for separating paper and low density materials from solid waste streams. The
large drag force to gravitational force ratio of the air on the low density materials causes materials such as
paper to separate from plastics and metals.
Magnetized
cylinder
Plastic and
metals input
Rotating belt
Ferrous
metals output
Plastic and
non-ferrous
metals output
Figure 11-3. Example of a magnetized separator. As the belt rotates around the magnetized cylindrical
roller the ferrous materials cling longer to the belt than non-ferrous materials fall off and thus allow
separation.
11-2
SOLIDS NOTES 11, George G. Chase, The University of Akron
Top View
Air Holes
Moving Belt
Monolayer of
Chopped plastic
Side View
Laser scan
to ID colors
by location
Moving Belt
Air jets use
puffs of air to
blow plastic
squares into
separate bins
Air Jets
Figure 11-4. Separation by color. Chopped plastic pieces are spread in a monolayer on a perforated
moving belt. Lasers scan the belt and identify the locations of plastic pieces of specific colors. Air jets
corresponding to the locations of the identified plastic pieces apply puffs of air to blow the plastic pieces
into bins.
10.2 Measuring Efficiency
Separation efficiency is directly related to processing costs. Most separations are not 100% efficient. The
outlet streams of the separation may be concentrated in the desired or undesired products, but may not be
pure. The challenge is to determine the best way to define and measure the efficiency of the separation.
The method of separation considered here is called the Grade Efficiency (For an alternative method based
on thermodynamic entropy, see the paper by Guistino et.al.2). A black box powder separation process is
shown in Figure 11-5. The total mass balance gives
M = Mc + M f .
The total separation efficiency,
(11-1)
ET , is defined as
ET =
Mf
Mc
= 1−
.
M
M
(11-2)
We assume that there is no agglomeration or communition in the separator. For a particle size, x, the
masses of size x in each stream are noted by M x , M fx , M cx . By analogy with Eq.(11-2), the grade
efficiency of separation of size x is defined as
Gx =
M cx
Mx
(11-3)
2
JM Giustino, G.G. Chase, and M.S. Willis, “Thermodynamic Separation Efficiency and Sedimentation
Criteria for Multiphase Processes: A Comparison of Rigorous and Approximate Models,” Separations
Technology, 5 (3), 153-164, 1995.
11-3
SOLIDS NOTES 11, George G. Chase, The University of Akron
Coarse Stream
Total mass Mc of
powder
Black box
Separation
Process
Feed Stream
Total mass M
of powder
Fines Stream
Total mass Mf
of powder
Figure 11-5. Black box (hypothetical) separation process to separation coarse particles from fine particles.
We know from the definitions of frequency distributions (see notes Section 3, Properties of Particulate
Solids) that the mass and mass components of each stream are related by
M x = M (fraction of size x)
= M f x dx
(11-4)
= M dFx
M cx = M c f cx dx = M c dFcx
similarly,
(11-5)
Hence, the grade efficiency is related to the size distribution functions by
G ( x) =
M c f cx
.
Mf x
(11-6)
A typical plot of the grade efficiency versus the particle size is shown in Figure 11-5. The area under the
curve plotted in Figure 11-6 represents the coarse cut (the stream with the larger particles) and the area
above the plotted curve represents the fines cut. Svrovsky1 gives more detail on defining and measuring
these curves.
1
Fines
Cut
Coarse
Cut
G (x)
0
x
Figure 11-6. Typical S-shaped grade efficiency curve. At a point, x, on the curve, G(x) represents the
fraction of particles of size x that are separated out of the feed stream and contained in the fines product
stream.
11-4
SOLIDS NOTES 11, George G. Chase, The University of Akron
For a continuous steady process the curve in Figure 11-6 is steady. For an unsteady process such as
filtration, the curve changes with time, as shown in Figure 11-7.
G (x)
Increasing
Time
G (x)
Increasing
Time
x
x
Cake Filtration
Depth Filtration
Figure 11-7. Comparison of typical grade efficiency curves for Cake Filtration and Depth Filtration. In a
typical cake filtration, as the filter cake depth increases, the cake itself improves the separation and the
grade efficiency shifts with increasing efficiencies for smaller particles. In Depth filtration, initially the
filter may perform very well. Gradually the capture sites in a typical depth filter are occupied (though other
mechanisms such as straining may occur) and the fine particles start to bleed through. Hence in Depth
Filtration the grade efficiency curve shifts toward larger particles as the filter becomes less efficient at
capturing small particles.
As an example of how you might apply the grade efficiency curve, consider the water filter pumps used by
backpackers on a hiking trail. One of the objectives of the filter is to remove harmful bacteria from water.
Bacteria are typically greater than 1 micron in size. If you have the choice of two filters with grade
efficiency curves shown in Figure 11-8, the savy backpacker would choose filter A because it is better than
99.99% effective at removing 1 micron particles. Filter B appears to be only about 50% effective at
removing 1 micron particles.
A
G (x)
B
1
10
x, microns
Figure 11-8. Comparison of grade efficiency curves for two filters, A and B. Filter A is approximately
99+% efficient at removing 1 micron particles. Filter B is only about 50% efficient at removing 1 micron
particles, though it is 99+% efficient at removing 10 micron particles.
11-5
SOLIDS NOTES 11, George G. Chase, The University of Akron
One of the reasons for inventing the Grade Efficiency is that it makes some calculations regarding the
separation of particles by size easier. Consider an arbitrary process in Figure 11-9a, where the amount of
material of size x entering the process is given by Mx and the amount of material of size x leaving in the
coarse and fine streams are given by
M cx = G x M x
(11-7)
M cx = (1 − G x )M x
(11-8)
You can use the grade efficiency to determine the amounts of material of size x in various streams of
processes that are cascaded in series or parallel. For example, in Figure 11-9b the streams exiting process 2
have amounts of size x given by
M c 2 x = G2 x G1x M x
(11-9)
M f 2 x = (1 − G 2 x )G1x M x
(11-10)
Suppose Mc2x is your product in stream C2 in Figure 11-9b. The amount of product in stream C1 is Mc1x =
G1xMx as given by Eq.(11-7). Because G < 1, then Mc2x < Mc1x. This can be generalized to conclude that
the more separations steps in your process the smaller the yield of your product.
Mcx
Mx
G(x)
Mfx
a. Arbitrary process that divides the feed stream M
into fines and coarse streams. The grade efficiency
of this process is given by G.
Mx
1
G1x
C1
2
G2x
C2
Mc2x
Mf2x
Mf1x
b. Two arbitrary process in series with grade
efficiencies G1 and G2 .
Figure 11-9. One and Two arbitrary processes in series with corresponding Grade Efficiencies.
11-6
SOLIDS NOTES 11, George G. Chase, The University of Akron
11.3 Cut Size and Sharpness of Cut
To compare efficiencies between steady state processes, we define cut size and sharpness of cut. Normally,
cut size refers to the 50% cut size, denoted x 50 . This is the particle size for which 50% of the particles exit
the separation process in the coarse product stream and 50% exit in the fines product stream.
Sharpness of cut is defined as a ratio of particle sizes specified at two efficiencies, typically at 20% and
80%. The sharpness of cut is defined as
I 80 / 20 =
x80
x 20
(11-11)
By this definition, and because the grade efficiency is a monotonically increasing curve, the sharpness must
be greater than or equal to unity. In an idealized case in which there is a perfect separation, where all
particles less than the 50% cut size exit in the fines stream and all particles greater than the cut size exit in
the coarse stream, the sharpness of cut equals unity. Real separation processes have a sharpness of cut
greater than unity. These concepts are shown in Figure 11-10.
Idealized
sharp cutoff
with
G (x)
G (x)
(a)
x80
=1
x20
(b)
80
50
The sharper the cut the
smaller the triangular areas
between the real and
idealized grade efficiency
curves.
50
20
x20 x50
x80
x50
x
x
Figure 11-10. (a)Typical grade efficiency curve with the particle sizes indicated for which the separation is
20, 50, and 80 percent efficient. (b) The idealized sharp cutoff grade efficiency curve is a vertical line
Other definitions of sharpness of cut are also used, including
x 20
which is the inverse of I 80 / 20 . x values such as 90/10 could also be defined.
x80
1.
I 20 / 80 =
2.
Variance in the slope,
3.
Slope of the grade efficiency curve, I x =
4.
Sum of triangular areas in Figure 11-10(b).
dG
.
dx
dG ( x )
dx
11-7
SOLIDS NOTES 11, George G. Chase, The University of Akron
11.4 Construction of the Grade Efficiency Curve
In the ideal case you would feed into your separator a material with a monodispersed particle size
distribution (of size x).
You would
•
Measure M x , M cx , M fx ,
•
Calculated G (x ) ,
•
Repeat for other size x (until you have enough points to construct your curve or you loose
patience).
This approach is not very realistic because of the difficulty in obtaining monodispersed materials
(especially in the very small particle size range) and because of the time and effort required.
In real applications you need to measure two of the following
•
Feed rate,
•
Coarse product rate,
•
Fines product rate,
and you need at least two of the following,
•
Feed particle size distribution,
•
Coarse particle size distribution,
•
Fine particle size distribution.
It is best to have data from the smaller of the two product streams because errors in sampling are smaller
than when you subtract two larger streams to get the smaller stream. As indicated in Figure 11-11, the flow
rate of the smaller stream may be less than the error of measurement of the larger streams.
10 lbm/hr
± ?
10,000 lbm/hr
± 50 lbm/hr
9,990 lbm/hr
± 50 lbm/hr
Figure 11-11. Example in which the size of the smaller product stream is smaller than the error in
measurements of the other two steams. Due to the error, it is better to measure the smaller stream directly
instead of calculating it from the two larger streams.
11-8
SOLIDS NOTES 11, George G. Chase, The University of Akron
11.5 Example 1
A sample of the feed, coarse, and fines streams for a separation of a material "Hexamethyl chicken wire" is
screened with the following results:
Screen
Size
(microns)
Average
Particle
Size
850
Feed Size
mass
Fraction
retained
on screen
Mass
Rate of
particles
size x in
Feed
Stream
Coarse
Size mass
Fraction
retained
on screen
Mass
Rate of
particles
size x in
Coarse
Stream
Fines Size
mass
Fraction
retained
on screen
Mass
Rate of
particles
size x in
Fines
Stream
∆Fx
Mx
∆Fcx
M cx
∆F fx
M fx
~0
~0
~0
600
725
0.30
0.45
0.075
425
512.5
0.40
0.45
0.325
300
362.5
0.20
0.09
0.365
212
256
0.10
0.01
0.235
1
1
1
Total
G(x ) =
M cx
Mx
The stream rates are:
•
Feed rate = 100 lbm/hr
•
Coarse Product Rate = 60
lbm/hr
0.9
•
Fines Product Rate = 40
lbm/hr
0.7
0.8
0.6
G(x)
Plot the grade efficiency curve and
calculate I20/80.
1
0.5
0.4
0.3
0.2
0.1
0
0
100 200 300 400 500 600 700 800 900 100
0
Size, microns
11-9
SOLIDS NOTES 11, George G. Chase, The University of Akron
SOLUTION
A sample of the feed, coarse, and fines streams for a separation of a material "Hexamethyl chicken wire" is
screened with the following results:
Screen
Size
(microns)
Average
Particle
Size
850
Feed Size
mass
Fraction
retained
on screen
Mass
Rate of
particles
size x in
Feed
Stream
Coarse
Size mass
Fraction
retained
on screen
Mass
Rate of
particles
size x in
Coarse
Stream
Fines Size
mass
Fraction
retained
on screen
Mass
Rate of
particles
size x in
Fines
Stream
∆Fx
Mx
∆Fcx
M cx
∆F fx
M fx
G(x ) =
~0
0
~0
0
~0
0
1
600
725
0.30
30
0.45
27
0.075
3
0.9
425
512.5
0.40
40
0.45
27
0.325
13
0.68
300
362.5
0.20
20
0.09
5.4
0.365
14.6
0.27
212
256
0.10
10
0.01
0.6
0.235
9.4
0.06
1
100
1
60
1
40
Total
The stream rates are:
•
Feed rate = 100 lbm/hr
•
Coarse Product Rate = 60 lbm/hr
•
Fines Product Rate = 40 lbm/hr
M cx
Mx
Note, mass retained on the 850 micron screen
is zero. In the limit Gx approaches 1 at this
size (Gx = 1 at the size for which all particles
exit the separator in the Coarse stream).
Plot Gx vs the Average particle size to
determine the x20, x50, and x80 values.
Plot the grade efficiency curve and calculate I20/80.
SOLUTION:
Recall that
1
M x = M ∆Fx
0.9
0.8
M cx = M c ∆Fcx
Using these equations and a basis of 1 hour, the
table is filled in.
0.6
G(x)
M fx = M f ∆F fx
0.7
0.5
0.4
0.3
The calculated points for the Grade Efficiency are
plotted on the graph and a curve is fitted to the
points. From the curve the x 20 , x50 , and x80
0.2
0.1
0
values are estimated to be 330, 450, and 625
microns respectively.
0
100 200 300 400 500 600 700 800 900 100
0
x
x
x
20
50
80
Size, microns
The sharpness of cut is calculated to be
330
= 0.528 .
625
BLANK
FORM FOR CALCULATING GRADE EFFICIENCY
I 20 / 80 =
11-10
SOLIDS NOTES 11, George G. Chase, The University of Akron
Screen
Size
(microns)
Average
Particle
Size
Feed Size
mass
Fraction
retained
on screen
Mass
Rate of
particles
size x in
Feed
Stream
Coarse
Size mass
Fraction
retained
on screen
Mass
Rate of
particles
size x in
Coarse
Stream
Fines Size
mass
Fraction
retained
on screen
Mass
Rate of
particles
size x in
Fines
Stream
∆Fx
Mx
∆Fcx
M cx
∆F fx
M fx
G(x ) =
M cx
Mx
The stream rates are:
•
Feed rate =
•
Coarse Product Rate =
•
Fines Product Rate =
1
0.9
Plot the grade efficiency curve and
calculate I20/80.
0.8
0.7
G(x)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
100 200 300 400 500 600 700 800 900 100
0
Size, microns
11-11
SOLIDS NOTES 11, George G. Chase, The University of Akron
11.5 Example 2
The grade efficiency represents the fractional amount of particles by mass of size x in the feed stream that
exits the separator in the coarse stream. Derive a formula in terms of Grade efficiencies for determining the
fractional amount of particles of size x in the coarse stream exiting separator 2 in the compound process
shown in Figure 11-12 where the grade efficiencies of both processes are the same (ie., G1x = G2x).
1
G1
M feed
Mc1 coarse
2
G2
Mf1 fines
Mc2 coarse
Mf2 fines
SOLUTION
The fractional amount of size x in the coarse stream exiting separator 2 can by determined by
(the subscript ‘x’ is dropped from this notation, but is implied).
For an arbitrary process, from Eqs.(11-7) and (11-8)
M c = GM
and
M f = (1 − G )M
M c1 = G1 M
and
M f 1 = (1 − G1 )M
(1)
and
M f 2 = (1 − G 2 )M f 1
(2)
Hence
For process 2
M c 2 = G2 M f 1
where M
f1
is the feed stream to process 2.
Hence, by combining equations (1) and (2) we get
M f 2 = (1 − G 2 )(1 − G1 )M
Since G1 = G2 this expression simplifies to
M f 2 = (1 − G ) M
2
or
M f2
M
= (1 − G ) .
2
11-12