SOLIDS NOTES 3, George G. Chase, The University of Akron
3. PROPERTIES OF PARTICULATE SOLIDS
Before we can discuss operations for handling and separating fluid/particle systems we
must understand the properties of the particles.
3.1 Individual particle characteristics
In your assigned reading is a discussion on the characterization of particles. The way that
we characterize the particles largely depends on the technique used to measure them.
The way that we measure a particle size is as important as the value of the
measured size. For example, how would you quantify yourself if measured by
1. Circumference around your waist?
2. Diameter of a sphere of the same displacement volume as your body?
3. Length of your longest chord (height)?
As you can deduce, the measured values have different meanings and will be
important relative to those meanings. If you are sizing a life jacket belt you would
interested in the first size. If you are buying a sleeping bag I suggest the last one.
Based on the measurement techniques the particle sizes are typically related to equivalent
sphere diameters by
a.
b.
c.
d.
The sphere of the same volume of the particle.
The sphere of the same surface area as the particle.
The sphere of the same surface area per unit volume.
The sphere of the same area when projected on a plane normal to the direction
of motion.
e. The sphere of the same projected area as viewed from above when lying in a
position of maximum stability (as with a microscope).
f. The sphere which will just pass through the same size of square aperture as
the particle (as on a screen).
g. The sphere with the same settling velocity as the particle in a specified fluid.
There are two other methods that I know of for sizing particles that are not based upon
comparison to a standard (sphere) shape.
a. The first method is to fit the particle area projected shape to a polynomial type
of relation. The values of the polynomial coefficients characterize the particle
shape.
b. The second method is through the use of Fractals. A fractal length can be
determined which characterizes the size of the particle and its dimensionality
somewhere between linear and two-dimensional.
We will not be spending any time with these latter two methods though they would be
interesting topics for a term paper. Sizes of common materials are listed in HANDOUT
3.1.
Probably among the earliest forms of particle classification (sizing) to be developed is
sieving. Several sieve standards exist which classify particles according to the size hole
through which the particles can pass. Class HANDOUT 3.2 lists the Tyler and the US
3-1
SOLIDS NOTES 3, George G. Chase, The University of Akron
standard mesh nominal sizes as well as the screen opening sizes in mm and inches. Also
in this handout is the Osmotics Inc. “Filtration Spectrum” which compares, among other
things, the relative sizes of common materials.
Except for the extreme case of long thin fibers, the particle mean size will be of the same
order of magnitude of the dimensions of the particle no matter which method is used.
There are a number of properties of particles that are of interest besides its size and
shape. Particles can repel or attract each other due to static charge build up, they are
affected by van der Waals forces (when they are small enough), they can stick,
agglomerate, break up, bounce off of each other, chemically react with each other, and
they are effected by the surrounding fluid phase due to drag an buoyant forces.
3.2 Measurements
There are a number of methods for measuring particle sizes and size distributions. Many
of these techniques are listed in HANDOUTS 3.3 and 3.4.
Some of these methods depend upon calibration with known particle sizes. A number of
suppliers now sell small spherical particles of nearly uniform size distributions for
calibration purposes.
Some of the more advanced methods of particle size measurement not only measure the
particle sizes but they will also provide the size distributions of the particles. One of the
better known instruments for this is the Coulter Counter. A brief description of the
electronic particle counter principle is given in HANDOUT 3.5.
For a given material, there are four types of particle size distributions that are possible:
(1) by number, (2) by length, (3) by surface, and (4) by mass (or volume).
Distributions can be reported either in terms of frequency (differential form) or by
cumulative (integral form) as shown below.
To explain how we mathematically represent the distribution data, lets suppose that you
measure the mass of particles by size by some unspecificed process. As an example your
measured data may be plotted as shown in Figure 3-1. You can normalize the plot by
dividing the masses of each size by the total mass, to obtain the mass fractions as shown
in Figure 3-2.
Finally, if we add the mass fractions cumulatively we get the Cumulative Mass Fraction
plot, shown in Figure 3-3.
3-2
SOLIDS NOTES 3, George G. Chase, The University of Akron
5
4.5
4
Mass, grams
3.5
3
2.5
2
1.5
1
0.5
0
1
2
3
4
5
Diameter, x, mm
Figure 3-1. Example mass quantities of an imaginary sample of particles.
1
0.9
0.8
Mass Fraction
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
Diameter, x, mm
Figure 3-2. Mass fractions from data in Figure 3-1.
3-3
SOLIDS NOTES 3, George G. Chase, The University of Akron
1
0.9
Cumulative Mass Fraction
0.8
0.7
Series5
0.6
Series4
Series3
0.5
Series2
0.4
Series1
0.3
0.2
0.1
0
1
2
3
4
5
Diameter, x, mm
Figure 3-3. Cumulative mass fraction plot of data from Figure 3-1.
From these Figures we see that the cumulatve mass fraction can be written
mathematically as
n
Fm ( xn )   Fm ( xi )
(3-1)
i 0
as a function of the nth particle size. Furthermore, we can write the increment in the
cumulative mass, Fm as
Fm ( xi ) 
Fm ( xi )
x
x
(3-2)
Fm
is the slope of the curve on the cumulative mass fraction plot. We define
x
this slope to be the frequency distribution of the mass fraction, f m , where
where
f m ( xi ) 
Fm ( xi ) dFm
.

x
dx
(3-3)
3-4
SOLIDS NOTES 3, George G. Chase, The University of Akron
Hence, we can relate the cumulative mass fraction to the frequency distribution by
n

Fm ( x n ) 
i 0
n
Fm ( xi )
x
x
f

i 0
m
( xi )x
.
(3-4)
xn
f

m
dx
0
Let the fractional amount of particles of size x be for any type of measurement (by mass,
number, area, etc.) be represented as
f ( x)  DISTRIBUTION FREQUENCY
(3-5)
(see L. Svarovsky, Solid-Liquid Separation, 3rd ed., Butterworth, London, 1990, chapter
2). If the particle size distribution is determined as the number fraction then the number
frequency distribution is given by
f N ( x)x 
Number of particles of size x
.
Total number of particles
(3-6)
where x is the differential range above and below size x that the number count
represents. If the particle size distribution is determined on a microscope by measuring
projected areas or by laser attenuation then the surface fraction or frequency distribution
based on surface area is
f s ( x)x 
Area of all particles of size x
.
Total area of all particles
(3-7)
Since f is a fractional amount, then integrating over all particle sizes gives the whole, or

 f ( x)dx  1
0
(3-8)
and if we integrate over only the range from zero to some size x we get the cumulative
fraction, F(x),
F ( x)   f ( x)dx
x
0
(3-9)
which is the area under the f(x) curve from 0 to x.
Plots of F and f have the general form
3-5
SOLIDS NOTES 3, George G. Chase, The University of Akron
1
F (x )
fo r F
f(x )
0
x
Figure 3-4. Typical f and F curves.
where f and F are also related by
f ( x) 
dF ( x )
dx
(3-10)
The frequency distributions, f(x), and the cumulative fraction, F(x), may be based on
numbers of particles or surface areas as described above, and are denoted with subscripts
N or S. Linear and volume (mass) basis for the distributions also exist and are denoted by
subscript L or M.
1. Number Distribution
fN(x)
2. Distribution by Length
fL(x)
3. Distribution by Surface
fS(x)
4. Distribution by Mass
fM(x)
(Not used in practice)
(Equivalent to distribution by volume)
The several types of distributions are all related to the number distribution by
f L ( x )  k1 x f N ( x )
(3-11)
f S ( x)  k 2 x 2 f N ( x)
(3-12)
f M ( x)  k 3 x 3 f N ( x)
(3-13)
where k1, k2, and k3 are geometric shape factors.
Similarly, the cumulative distributions can be related
FL ( x)   k1 x f N ( x)dx   k1 x dFN   k1 xf N x
(3-14)
FS ( x)   k2 x 2 f N ( x)dx   k2 x 2 dFN   k2 x 2 f N x
(3-15)
FM ( x)   k3 x3 f N ( x)dx   k3 x3 dFN   k3 x3 f N x .
(3-16)
x
x
0
0
x
x
0
0
x
x
0
0
3-6
SOLIDS NOTES 3, George G. Chase, The University of Akron
Often, experimental data are reported in discrete form (such as from a sieve analysis).
For these data it is easier to work with discrete forms of the integral equations:
i
FN ( xi )   f N ( x j )x j
(3-17)
j 1
fN (x j ) 
where
nj
(3-18)
Nx j
where n j is the number of particles in the jth set, N is the total number of particles, and
x j .is the size increment range that n j represents.
As an example, to find k2, we start with
f s x 
n j Aj
n A
j
Nxf Nj Aj

 Nxf
j
Nj
Aj

f Nj Aj
f
Nj
Aj
(assuming constant x ).
Let Aj  x 2j , and combine with Eq. (3-12), upon rearrangement we get
k2 
1
.
 x f x
2
j Nj
There are several equations that are typically fitted to the distribution. The most widely
used function is called the log-normal distribution. It is a two-parameter function that
gives a curve, which is skewed to the left compared to the familiar bell curve. This
function is normally used because in most cases there are many more measured fine
particles than larger particles.
The lognormal function is best described first by considering the normal distribution of
the Gaussian (bell shaped) curve shown in Figure 3-5a:
  x  x2 
dF
1


exp 
2


dx  2
2



(3-19)
where F is the cumulative undersize fraction of particles, x is the particle size,  is the
standard deviation, and x is the mean particle size.
To fit Eq.(3-19) to experimental data (such as from a sieve analysis) first make any
adjustment necessary for left or right bias (that is, use the diameters associated with the
center of each bin and not the left or right edges). The average diameter and standard
deviations are determined from
x
x n
i
N
i
  xi Fi
(3-19a)
3-7
SOLIDS NOTES 3, George G. Chase, The University of Akron

 n x
i
 x
2
i
N 1
 N 
2
 
 Fi xi  x 
 N 1
(3-19b)
To obtain the log-normal distribution, Figure 3-5b, we substitute ln(x) for x and ln(g)
for . This gives

 ln x  ln x
xdF
1
g


exp 
2
dx
2 ln  g
ln  g 2



2




(3-20)
where x g is the geometric mean and is equal to the median size (where 50% of the
particles are greater in size and 50% are smaller in size).
Figure 3-5a Normal Gaussian curve.
Figure 3-5b. Log-normal curve.
To fit Eq.(3-20) use the following expressions:
ln(x g ) 
 n ln(x ) 
i
i
N
 n ln(x )  ln(x )
2
ln( g ) 
i
i
N 1
 ln(x )F
i
i
 N 
2
 
 Fi ln(xi / x )
 N 1
(3-20a)
(3-20b)
3-8
SOLIDS NOTES 3, George G. Chase, The University of Akron
When Eq.(3-20) is rearranged and the substitution
ln xm  ln xg  ln 2  g
(3-21)
is applied, we get the more convenient form
 ln x  ln x  2 
dF
1
m
2


exp  ln  g / 2 exp 
2


dx xm ln  g 2
2
ln

g




(3-22)
in which xm represents the mode because it is the size at which dF/dx has its maximum
(recall f(x) = dF/dx, hence f is maximum at its mode, at xm). Svarovsky (L. Svarovsky,
Powder Technology, 7, 351-352, 1973) recommends writing Eq. (3-22) as

 x 
dF
 a exp   b ln 2   
dx
 xm  

(3-22a)
where
1
a
xm
b
b
 
 
1/ 2
  1
exp  
 4b 
(3-22b)
1
2 ln 2  g
(3-22c)
to simplify the calculations.
1
0.9
0.8
Sigma = 1.5
0.7
Sigma = 3
f(x)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
30
-0.1
X
Figure 3-6. Comparison between log normal curves with  g  1.5 and  g  3 . Both
curves have the same area, but the larger standard deviation causes the second curve to
have a smaller peak and more spread.
3-9
SOLIDS NOTES 3, George G. Chase, The University of Akron
(HANDOUT 3.6)
A sample of M&M’s ™
with peanuts are weighed
as listed in Table 3-1.
Using an average density
of 1.23 grams per cubic
centimeter, the average
candy diameter (assuming
spherical shape) is
calculated. Plot the
frequency distribution and
the cumulative frequency
distribution of the average
diameter of the candies.
Solution: to make the
desired plots, the data
points must first be
organized into bins with
specified size increments.
The
size of the bins is
set at 0.05 cm. All M&Ms
of size less than 1.5 cm are
placed in the first bin. All
sizes that fall between 1.5
and 1.55 cm are placed in
the second bin, and so on.
The values for nj are
determined by counting the
number of M&Ms that fall in
a given size increment (bin
size) and are assigned to the
average size in the
increment.
The results are plotted in
Figure 3-7a.
Table 3-1a. Mass and diameter distribution of M&M’s.
Grams Dia, cm Size < Avg size No.
2.06
1.473
1.5
1.475
1
2.18
1.501
2.18
1.501
2.21
1.508
2.22
1.511
2.35
1.540
2.36
1.542
2.37
1.544 1.55
1.525
7
2.4
1.550
2.42
1.555
2.47
1.565
2.49
1.570
2.53
1.578
2.57
1.586
2.58
1.588
2.59
1.590
2.63
1.598
1.6
1.575
9
2.71
1.614 1.65
1.625
1
2.94
1.659
2.99
1.668
1.7
1.675
2
3.01
1.672 1.75
1.725
1
Total
fdx
f
F
0.047619 0.952381 0.047619
0.333333 6.666667 0.380952
0.428571 8.571429 0.809524
0.047619 0.952381 0.857143
0.095238 1.904762 0.952381
0.047619 0.952381
1
21
1
Frequency Distribution of M&Ms
Frequency Distribution
EXAMPLE 3-1
10
9
8
7
6
5
4
3
2
1
0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.4
1.5
1.6
1.7
1.8
Diameter, cm
Figure 3-7a. Plot of frequency and cumulative frequency
distributions for M&M’s.
3-10
f
F
SOLIDS NOTES 3, George G. Chase, The University of Akron
Alternatively, the data can be fitted to a distribution curve, the Gaussian Distribution
curves in this case. Using Eqs.(3-19a) and (3-19b) the mean and standard deviation are
calculated, from which the frequency and cumulative frequency distributions are
calculated.
The calculated mean is 1.567 cm and the standard deviation is 0.0551. The frequency
distribution is calculated with Eq.(3-19) and the cumulative distribution is calculated
using Eq.(3-9) and trapezoidal rule. The data points for creating the plots from the
equations (not from the experimental data) are listed in Table 3-1b and the results are
plotted in Figure 3-7b.
Table 3.1b. M&M fitted distribution curves.
x
1.45
1.46
1.47
1.48
1.49
1.5
1.51
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.6
1.61
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.7
1.71
1.72
1.73
1.74
1.75
arg=(-(x-xavg)^2)/2/stdev^2
-2.2623
-1.8930
-1.5566
-1.2530
-0.9823
-0.7446
-0.5397
-0.3677
-0.2286
-0.1224
-0.0491
-0.0087
-0.0012
-0.0266
-0.0849
-0.1760
-0.3001
-0.4570
-0.6468
-0.8696
-1.1252
-1.4137
-1.7351
-2.0894
-2.4766
-2.8967
-3.3497
-3.8355
-4.3543
-4.9059
-5.4905
exp(arg)
0.1041
0.1506
0.2109
0.2856
0.3744
0.4749
0.5829
0.6923
0.7956
0.8848
0.9520
0.9913
0.9988
0.9738
0.9186
0.8386
0.7408
0.6332
0.5237
0.4191
0.3246
0.2432
0.1764
0.1238
0.0840
0.0552
0.0351
0.0216
0.0129
0.0074
0.0041
f(eq.3-19)
0.7533
1.0898
1.5257
2.0667
2.7091
3.4363
4.2176
5.0091
5.7565
6.4015
6.8884
7.1724
7.2266
7.0455
6.6467
6.0676
5.3597
4.5812
3.7891
3.0325
2.3485
1.7599
1.2762
0.8954
0.6080
0.3994
0.2539
0.1562
0.0930
0.0536
0.0299
dx
dF=0.5(f1+f2)dx
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.0092
0.0131
0.0180
0.0239
0.0307
0.0383
0.0461
0.0538
0.0608
0.0664
0.0703
0.0720
0.0714
0.0685
0.0636
0.0571
0.0497
0.0419
0.0341
0.0269
0.0205
0.0152
0.0109
0.0075
0.0050
0.0033
0.0021
0.0012
0.0007
0.0004
F =sum(dF)
0
0.0092
0.0223
0.0403
0.0641
0.0949
0.1331
0.1793
0.2331
0.2939
0.3603
0.4306
0.5026
0.5740
0.6425
0.7060
0.7632
0.8129
0.8547
0.8888
0.9157
0.9363
0.9515
0.9623
0.9698
0.9749
0.9781
0.9802
0.9814
0.9822
0.9826
3-11
SOLIDS NOTES 3, George G. Chase, The University of Akron
10.0000
1
8.0000
0.8
6.0000
0.6
f
4.0000
0.4
2.0000
0.2
0.0000
F
0
1.45
1.5
1.55
1.6
1.65
1.7
1.75
Figure 3-7b. Frequency and cumulative distributions of the M&Ms as determined from
the fitted Gaussian function.
Comparison between Figures 3-7a and 3-7b show subtle differences. In Figure 3-7a the
frequency distribution shows two modes (bi-modal distribution) indicating that the
M&Ms tend to be near two different sizes. The fitted Gaussian curve in Figure 3-7b only
shows on mode. This is because the fitted Gaussian curve by design only has one mode.
3-12
SOLIDS NOTES 3, George G. Chase, The University of Akron
3.3 Choice of Mean Particle Size
As shown in handout 3 and the previous discussions, there is a bewildering number of
different definitions of "mean" size for a particle. The choice of the most appropriate
mean is vital in most applications.
As can be seen in Figure 3.8, two different size distributions may have the same
arithmetic mean, but all of the other means may be different. (HANDOUT 3.7)
MODE
HARMONIC MEAN
ARITHMETIC MEAN
MEDIAN
f
QUADRATIC MEAN
CUBIC MEAN
f
x
Figure 3.8 Comparison of size distributions.
The mode is the x value at which f(x) is a maximum. The median is the x value at which
F(x) = 0.50.
3-13
SOLIDS NOTES 3, George G. Chase, The University of Akron
The various means are defined by:

1
g x   g ( x)dF
(3-23)
0
or by the equivalent expressions
1
g ( x)   g ( x) f ( x)dx   g ( x) f ( x)x
(3-24)
0
g(x) =
NAME OF MEAN
x
ARITHMETIC MEAN, x a
x2
QUADRATIC MEAN,
x3
CUBIC MEAN, x c
log x
GEOMETRIC MEAN,
1/x
xq
xg
HARMONIC MEAN, x h
Example, suppose we want the cubic mean of a set of particles for which
we know the number distribution. The mean is defined such that
x c N   xi3 ni ,
3
or
x c   xi3
3
ni
  xi3 f Ni x
N
where f N x 
ni
N
hence
xc  3
x
3
i
f Ni x
Suppose you have the mass distribution frequency of a set of particles and
you want the geometric mean. How would you calculate the geometric
mean from the given mass distribution frequency?
log( x g )   log( xi ) f M ( xi )x
hence
x g  10 
log(xi ) f M ( xi ) x
3-14
SOLIDS NOTES 3, George G. Chase, The University of Akron
The mean particle size is rarely quoted in isolation. It is usually related to some
measurement technique and application and used as a single number to represent the full
size distribution. The mean represents the particle size distribution by some property
which is vital to the application or process under study. If two size distributions have the
same mean (as measured using the same methods) then the behavior of the two materials
are likely to behave in the process in the same way.
It is the application therefore which governs the selection of the most appropriate mean.
Usually enough is known about a process to identify some fundamentals, which can be
used as a starting point. The fundamental relations may be overly simple to describe the
process fully, but it is better than randomly selecting mean definition.
EXAMPLE 3-2. Comparison of mass versus number count.
Consider measuring the size distribution by sieving. The results of a sieve analysis may
give the size distribution as (HANDOUT 3.8)
Table 3.2 Sieve analysis of a sample of particles. Mass, number, and area fractions are
calculated.
Sieve analysis of a sample of particles. Mass, number and
area fractions are calculated.
Note 1
Note 2
AVG
VOLUME
SIEVE
SIZE SIEVE MASS
ON VOLUME
V1
SIZE,
MASS,
TRAY,
MM SIZE, MM
g FRAC
MM^3
FRAC MM^3
pan
0
0.04
0.05
0.10 0.03
38.46
0.03
0.00
0.06
0.08
0.40 0.11 153.85
0.11
0.00
0.10
0.14
0.70 0.19 269.23
0.19
0.01
0.18
0.24
0.90 0.25 346.15
0.25
0.06
0.30
0.36
0.70 0.19 269.23
0.19
0.21
0.42
0.50
0.50 0.14 192.31
0.14
0.60
0.59
0.71
0.20 0.06
76.92
0.06
1.69
0.83
0.92
0.10 0.03
38.46
0.03
3.63
1.00
TOTAL MASS
3.60
1.00 1384.62
1.00
A1
AREA
TRAY AREA
FRAC MM^2
MM^2 FRAC
NUMBER NUMBER
67293.01
58141.16
21045.58
5660.10
1266.29
320.67
45.42
10.60
0.44
0.38
0.14
0.04
0.01
0.00
0.00
0.00
0.01 518.00
0.02 1243.20
0.06 1286.65
0.17 982.00
0.40 504.18
0.79 254.88
1.59
72.13
2.64
27.98
0.11
0.25
0.26
0.20
0.10
0.05
0.01
0.01
153782.82
1.00
4889.01
1.00
The mass fraction is found simply by dividing the sample masses (sieve mass) by the sum
of the masses. Dividing the sample mass by the particle intrinsic density (assumed here
to be 2.6 g/cm3) gives the volume of the particles in the sample. Dividing the sample
volume by the volume of one particle ( 43  R 3 ) where R is the sieve size opening, gives
the number of particles for that sample. The total surface area of the particles of a given
size is obtained by multiplying the number of particles times the surface area of one
3-15
SOLIDS NOTES 3, George G. Chase, The University of Akron
particle ( 4 R 2 ). The number and area fractions are found by dividing the sample values
by the totals.
The plot in Figure 3.9 shows that the modes of the three distributions vary widely. The
number distribution and surface area distribution are skewed greatly to the small particle
size. This shows that a small mass of the fines contains a large number of particles.
A property such as turbidity is sensitive to the total number of particles, hence the large
number of fines will cause the fluid to be cloudy. A process such as filtration is sensitive
to the total surface area of the particles due to the drag resistance to flow across the
surface.
0.50
0.45
0.40
Fraction
0.35
0.30
Mass & Volume Frac
Number Frac
0.25
Area Frac
0.20
0.15
0.10
0.05
0.00
0.00
0.20
0.40
0.60
0.80
1.00
Avg Particle Size, mm
Figure 3.9. (a) Comparison of the fractional distributions of the particle size
distributions.
EXAMPLE 3-2 Continued. Using the given data we can calculate the frequency
distributions fm, fn, and fs using equations 3-10, 3-12, and 3-13. The constants k2 and k3
are calculated to provide the conversions.
3-16
SOLIDS NOTES 3, George G. Chase, The University of Akron
(top
tray)
(pan)
SIEVE
RETAINED
SIZE
MASS
x
Avg
size
fm =
fn =
fmdx/dx
fm/k3 x^3
mm
g
mm
1.000
0.000
0.830
0.100
0.915
0.170
0.028
0.590
0.200
0.710
0.240
0.056
0.036
0.163
0.000
0.000
0.034
0.155
0.231
0.001
0.000
0.062
0.420
0.500
0.505
0.170
0.139
1.078
0.817
0.012
0.001
0.305
0.300
0.700
0.360
0.120
0.194
4.168
1.620
0.065
0.001
0.849
0.180
0.900
0.240
0.120
0.250
18.084
2.083
0.282
0.002
1.638
0.100
0.700
0.140
0.080
0.194
70.862
2.431
1.660
0.003
3.276
0.060
0.400
0.080
0.040
0.111
217.014
2.778
10.167
0.003
6.553
0.040
0.100
0.050
0.020
0.028
222.222
1.389
20.822
0.001
5.242
0.000
0.000
0.020
0.040
0.000
0.000
0.000
0.000
0.000
0.000
Total mass
dx
dFm=fmdx
fmdx/x^3
(mass frac)
k3=sum(fmdx/x^3)
3.600
x^2fndx
fs =
k2x^2fn
k2=1/sum(x^2fndx)
533.620
100.707
25.000
fm,fn,fs
20.000
15.000
fn
fn
10.000
fs
5.000
0.000
0.000
0.200
0.400
0.600
x, avg particle size
0.800
1.000
Figure 3.9b. Plot of mass, number, and area frequency distributions.
3-17
SOLIDS NOTES 3, George G. Chase, The University of Akron
EXAMPLE 3-3. Application: cake filtration, cake washing, dewatering, flow through
packed beds and porous media.
If the particle size distribution is known, what definition of the mean
should be used?
In flows through a packed bed we can consider the pores to be conduits.
We can apply the concept of a friction factor and a Reynolds number.
Since the geometry of an arbitrary pore is not cylindrical, we apply the
hydraulic radius, Rh.
cross - section area available for flow
wetted perimeter
volume available for flow

total wetted surface
 volume of voids 


 volume of bed 

 wetted surface 


 volume of bed 
Rh 

(3-25)

a
where  is the bed porosity and a is a surface area. This surface area is
related to the specific surface area, a s , of the solids (total particle
surface/volume of particles) by
a  a s 1   
(3-26)
The specific surface area in turn is related to the mean particle diameter
(assuming the particle can be represented by a sphere)
x  Dp 
6 6  total volume of particles

as
total surface of particles
(3-27)
For spheres the total volume of particles is given by
4
total volume   R 3
3
1
  D pi D pi2
6
1
  xi S i
6


(3-28)
and the total surface area of the particles is given by
total surface area   4 Ri    Di   Si
2
2
(3-29)
Hence, we get the mean particle diameter to be
3-18
SOLIDS NOTES 3, George G. Chase, The University of Akron
x S
S
  xdF
x
i
i
i
where
1
Si
S
 dFs
(3-30)
i
s
0
where the latter expression is the analytical formulation.
This latter expression defines the mean to be the arithmetic mean,
x   x a  s (from Eq. 3-23) of the distribution by surface.
Next, we must relate this to a size distribution by mass (the usual way of
measurement). The surface distributions by surface and mass can be
related by
f m  x   kxf s  x 
(3-31)
or
dFs 
dFm
kx
where k is a constant that accounts for the geometric shape of the particles.
It is assumed here that k is independent of x.
Since the mean size is given in Eq. (3-30), then combining (3-30) and (331) we get
x 
1
a s
1
1
  dFm 
k0
k
(3-32)
where the integral is unity. If we go back to Eq.3-31, we can integrate to
obtain
1
1
0
0
 dFs  
dFm
kx
(3-33)
or, since k is not a function of x,
1
k
0
dFm
x
(3-34)
where the RHS of Eq.(3-34) is the definition for the Harmonic mean,
1
 xh  m , of the mass distribution given by Eq. (3-23). Hence this shows
that the surface arithmetic mean is equal to the mass harmonic mean,
 xa  s   xh  m .
Therefore, for flow through packed beds, filter cakes, etc., the appropriate
mean particle size definition is the arithmetic average of the surface
3-19
SOLIDS NOTES 3, George G. Chase, The University of Akron
distribution. This is shown to be equivalent to the mass distribution
harmonic mean.
EXAMPLE: 3-4. Mass recovery of solids in a dynamic separator such as a gravity
settling tank.
For a settling process in which mass recovery is to be optimized, which
would be the most appropriate mean particle size?
Total recovery of any separator can be obtained by combining the feed
size cumulative distribution, F(x), with the operating grade efficiency
curve, G(x). Mathematically, this is written as
1
ET   G ( x)dF
(3-35)
0
where Et is the recovery by mass.
A simple plug flow model of the separation in a settling tank without
flocculation gives the grade efficiency in the form
G( x) 
ut A
Q
(3-36)
where A is the settling area, Q is the suspension flow rate, and ut is the
terminal velocity of particle size x.
Assuming Stoke's law for the terminal velocity
ut 
x 2  g
18
(3-37)
then these three equations can be combined to obtain
A g
 x 2 dF
18Q 0
1
Et 
(3-38)
where the integral defines the quadratic mean of the particle size
distribution by mass.
We will discuss Grade Efficiency in further detail in a later section when we discuss
separations processes.
3-20
SOLIDS NOTES 3, George G. Chase, The University of Akron
3.4 Drag Force on a Spherical Particle
Probably the most significant force acting on particles in a fluid-particle medium is the
drag force due to the relative motion between the fluid and the particles. A summary of
the derivation of the governing equations is
given here.
Fb = m f g
From a free body diagram, Figure 3.10, we
can write a balance of forces acting on a
spherical particle. The balance of forces
shows that the accelerating force acting on the
u
PAR T CI LE
particle is given by
Fk
M O VEM EN T
Fa  Fg  Fb  Fk
(3-39)
D RI EC T OI N
Initially, when a particle falls through a fluid
the particle velocity accelerates. After a short
distance the particle reaches its terminal
velocity and its acceleration goes to zero.
This means that the force of acceleration, Fa is
zero.
Hence, at terminal velocity the kinetic force
acting on the particle is given by
Fk  Fg  Fb
F g = m pg
Figure 3-10. Free body diagram on particle of
diameter R.
(3-40)
In Figure 3.10 mp is the mass of the particle and mf is the mass of the displaced fluid with
the same volume as that of the particle. These masses are equal to the volume of the
particle times the respective particle or fluid densities. The kinetic force becomes

Fk  43  R 3 g  p  

(3-41)
We define the drag coefficient, Cd, by the expression
Fk  Cd A KE
(3-42)
where A is the projected area normal to the flow and KE is the characteristic kinetic
energy. When we substitute in the projected area of a sphere, R2, and the kinetic
energy, 1/2 u2, into Eq. (3-42) then we can derive a working equation for determining
the drag coefficient as
C D  83 Rg

p

u
2

(3-43)
In order to use this expression to determine values for CD we must run experiments. The
experiments may be in the laboratory or they may be thought experiments for limiting
case solutions.
Lets consider the limiting case of creeping flow around the sphere as shown in Figure
3.8. This operation is discussed in some detail by Bird et.al. (1960).
3-21
SOLIDS NOTES 3, George G. Chase, The University of Akron
Z
r

z

rs
ni (
Y
)
x
y
X
v
Figure 3-11. Flow around a sphere of radius R. The flow is in the positive
z-direction such that there is symmetry in the -direction. At distances far
from the sphere the flow velocity is uniform at a value v. This problem
is equivalent to the particle falling in the negative z-direction through a
stationary fluid.
For creeping flow the dominant term in the momentum balance is the viscous force term,
which at the continuum scale gives
   0
(3-44)
where the stress tensor is related to the velocity by the Newtonian Fluid model. Since the
fluid motion around the sphere varies in the r- and -directions, it is mathematically
easier to solve the resulting differential equations in terms of the stream function, .
The stream function is related to the velocities in spherical coordinates by:
vr  
1 
r sin  
(3-45)
v  
1 
r sin   r
(3-46)
2
In terms of the stream function, the momentum balance in spherical coordinates becomes:
2
  sin    1   

   0
  2
r   sin    
 r
(3-47)
where the [ ] term is a differential operator and where symmetry is assumed in the 
direction (hence no dependence on .
Equation (3-47) is solved with the boundary conditions
3-22
SOLIDS NOTES 3, George G. Chase, The University of Akron
vr  
1 
0
r sin  
at r = R
(3-48)
v  
1 
0
r sin   r
at r = R
(3-49)
for r  
(3-50)
2
   21 v r 2 sin 2 
The first two boundary conditions mathematically describe the contact of the fluid to the
sphere surface. The third boundary condition shows that at distances far from the sphere
the velocity becomes v.
The last boundary condition suggests that
  f (r ) sin 2 
(3-51)
When this function is substituted into Eq.(3-47) we get the linear, homogeneous fourthorder equation
 d2
2  d 2
2
 2  2   2  2  f (r )  0
r   dr
r 
 dr
(3-52)
Assuming a solution of the form f (r )  Cr n shows that n may have the values of 1,1,2,4 hence we get the functional form for f(r) as
f (r ) 
A
 Br  Cr 2  Dr 4
r
(3-53)
where A,B,C and D are constants.
Applying the boundary conditions and the definitions for the stream function (Eqs. (345)-(3-46) and (3-48)-(3-50)) gives the velocity profiles
3
vr  3 R 1  R  
 1  2  2    cos
v 
r
 r  
(3-54)
 3 R 1  R 3 
v
  1  4  4    sin 
v
r
 r  

(3-55)
We could derive an expression for the kinetic force on the sphere by using the
momentum balances and an expression for the pressure distribution. A more direct way
is to recognize that the drag force on the sphere is directly related to the viscous
dissipation. For a Newtonian fluid, we can evaluate the kinetic or drag force directly
using
2  
v  Fk        : v r 2 dr sin  d d
(3-56)
0 0 R
Insertion of Newton’s Law of viscosity for the stress tensor in Eq.(3.41) and substitution
of the velocity profiles in Eqs.(3.39) and (3.40) yields the kinetic force as
3-23
SOLIDS NOTES 3, George G. Chase, The University of Akron
Fk  6  v R
(3-57)
This expression is known as Stoke's Law. Defining the Reynold's number as
Re 
d p v 
(3-58)

where d p is the particle diameter, we seek a correlation to relate the drag force to the
Reynold's number. A correlation would allow us to extend our applications to flow
conditions in which the creeping flow solution does not apply.
The drag coefficient, Cd, defined by expression (3-42) may be combined with Eqs.(3-57)
and (3-58) to derive
Cd 
24
Rep
(3-59)
which is the Stoke's Law condition for the drag coefficient and holds for Rep less than
one. For larger Reynolds numbers we need to use correlations obtained from
experiments. A number of references give the familiar drag coefficient correlation as
shown in Figure 3.12 for flow around spheres. (HANDOUT 3.9)
100000
Expl curve
10000
Stokes
Intermediate
Cd
1000
Newton Law
100
10
1
0.1
0.001
0.01
0.1
1
10
100
1000
10000
100000
Re
Figure 3.12. Drag coefficient for spheres versus Reynolds number. Three approximate
curves are overlayed onto the experimental curve. The approximate curves are, from left
to right, CD  24 / Rep (Stoke’s Law range for Rep<1), CD  18.5 / Rep3/5 (Intermediate
range for 1<Rep<1000), and CD  0.44 (Newton’s Law range 1000<Rep<100,000).
3-24
SOLIDS NOTES 3, George G. Chase, The University of Akron
For Reynold’s numbers less than 1 Stokes Law applies and this is known as Stoke’s Law
range. For Reynolds numbers greater than about 1000 and less than 105, where CD is a
constant, this is known and Newton’s Law range. Between these two ranges is known as
the intermediate range.
As can be seen in Figure 3.4 of the text by Coulson and Richardson (Chemical
Engineering, Vol. 2, 4th ed, Pergamon, 1991), above Rep of about 105 there is a sudden
decrease in the drag coefficient. In the book notation Re’=Rep and 2Cd=R’/u2. Rep >105
we get Cd=0.08.
Figures 3.2 and 3.3 in Coulson and Richardson (ibid) show the transition from smooth,
well behaved laminar flow (Stoke’s regime), into the turbulent ranges and the formation
of fluid eddies as the boundary layer separates from the particle surface. At the highest
flow range new mechanisms can become important as the fluid separates away from the
particle surface and cause the observed decrease in the drag coefficient.
If we rearrange Eq.(3-43) we can solve for the terminal velocity of the particle to be
ut 
4
3
d p g  p   


CD   
(3-60)
which applies to all flow regimes. When we substitute in Stoke’s Law, Eq.(3-60), we get
the terminal velocity to be
ut 

gd p2  p  

(3-61)
18
in Stoke’s Law range. Similarly in Newton’s Law range substitution of CD = 0.44 yields
ut  173
.

d p g p  


(3-62)
Literature references have other correlations for representing these various ranges of
Reynolds numbers.
These correlations only relate the motion to a few of the important factors (density, size,
Reynold’s number). There are many other factors that may become significant in given
situations. These include








proximity to vessel walls
particle surface roughness
particle shape
Brownian motion (for dp < 1 m)
external forces (electrical current, magnetic fields)
sound waves
rigid vs. deformable particles (ie., droplets)
particle concentration
The last topic in the list will be discussed further in a later section.
3-25
SOLIDS NOTES 3, George G. Chase, The University of Akron
3.5 Drag Force on Non-Spherical Particles
The shape and orientation of the particle has an important effect on the flow profiles
around the particle. McCabe and Smith (Unit Operations of Chemical Engineering, 6th
ed, McGraw-Hill, N.Y., 2001), Figure 7.3, and Perry’s Chemical Engineer’s Handbook,
(6th ed., McGraw-Hill, N.Y., 1984) Figure 5-76) show the correlation for the drag
coefficients for spheres, disks, and cylinders.
It is not practical to try to derive correlations for all particle shapes and orientations,
especially when in the chemical process industry particles in settling operations tumble
and rotate.
Kunii and Levenspeil studied this problem and developed a correlation based upon
sphericity (1966).
Sphericity is a measure of how close a particle is to being a sphere defined as

surface area of a sphere with same volume as the particle
actual surface area of the particle
(3-63)
The sphericity of some common materials are given in Table 3.3. (HANDOUT 3.10).
Table 3-3 Sphericity of Some Common Materials (McCabe & Smith, 6th ed, pg945;
Perry’s Handbook 6th ed, pg 5-54).
PARTICLE MATERIAL
Sphere
Cube
Short Cylinder (Length=Diameter)
Berl saddles
Raschig rings
Coal dust, natural (up to 3/8 inch)
Glass, crushed
Mica flakes
Sand
Average for various types
Flint sand, jagged
Sand, rounded
Wilcox sand, jagged
Most crushed materials
SPHERICITY
1.0
0.81
0.87
0.3
0.3
0.65
0.65
0.28
0.75
0.65
0.83
0.6
0.6 to 0.8
Kunii and Levenspiel (Fluidization Engineering, John Wiley, N.Y. 1969, pg 77) took data
from Brown (G.G. Brown et.al., Unit Operations, John Wiley, N.Y., 1950) and calculated
the relationships plotted in Figure 3-10, HANDOUT 3.11, relating Cd to Rep.
It turns out that the product Cd Rep 2 is independent of velocity, which makes it convenient
for calculations. Using Eq.(3-43) and the definition of the Reynold’s Number we get
3-26
SOLIDS NOTES 3, George G. Chase, The University of Akron
Cd Rep
2
 p     d pu 

  83 Rg
u 2   

gd 3p   p   
 43
2
 43 N GA
NGA 
where
2
(3-64)

gd 3   s   
(3-65)
2
is known as the Galileo number. With this chart and the correlation in Eq. (3-64) the
terminal velocity can be calculated from the material properties and the sphericity.
1.E+10
1.E+09
1.E+08
1.E+07
Plot to determine drag
coefficients of irregularly
shaped particles at terminal
velocity. The particles are
randomly oriented relative to
the flow direction. Shape is
accounted for by the
sphericity.
SPHERICITY
0.2
0.4
0.6
0.8
1.0
CdRep^2
1.E+06
1.E+05
1.E+04
1.E+03
1.E+02
1.E+01
1.E+00
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
Rep
Figure 3-13. Drag coefficient – Reynolds number relationship for non-spherical
particles. Equations 3-54, 3-60, and 3-61 are used with this chart. The particle diameter
is the volume equivalent diameter, xv, of the sphere with the same volume as the particle.
Haider and Levenspeil (Powder Technology, 58, 63, 1989) also found a useful
relationship for direct evaluation of terminal velocity of particles. The correlation is
shown in Figure 3-14 (HANDOUT 3.12) where a curve fit of the plot gives
3-27
SOLIDS NOTES 3, George G. Chase, The University of Akron
 18 2.335  1744
. 
ut*   *2 

d p*0.5
 d p

1
for 0.5    10
.
(3-66)
and the dimensionless velocity and particle diameter are defined as


2
*


ut  ut
    g
p






1/ 3
(3-67)
1/ 3
    g
p
*
 .
dp  dp 


2


and
(3-68)
Sphericity =
100
ut*
10
1.0
0.9
...
0.5
0.23
0.123
0.043
0.026
Sphericity for
Disks only
1
0.1
0.01
1
10
100
1000
10000
dp*
Figure 3-14. Plot of data taken from Kunii and Levenspiel, Fluidization Engineering, 2nd,
Butterworth, Boston, 1991. Dimensionless terminal velocity and particle diameter are
defined in Eqs.(3-67) and (3-68).
3-28
SOLIDS NOTES 3, George G. Chase, The University of Akron
EXAMPLE 3-5. Compare the terminal velocity of a cube of titanium, 5mm on each side,
falling through maple syrup and falling through water. Properties: titanium density =
7.14 g/cc; syrup density = 0.95 g/cc, syrup viscosity = 3000 cP; water density = 1.0 g/cc,
water viscosity = 1.0 cP.
SOLUTION:
For a cube   081
. . Since settling depends upon mass average of the particle
size, then the appropriate diameter is that of a sphere of the same volume.
3
m 

9 3
volume  l  (5mm) 
  125 x10 m
1000
mm


3
dp  3
6

3
125x10 9 m3  6.20 x10 3 m
For the syrup:
d p*
ut*
2
6


 kg   100cm 
2
 (0.95g / cc) 7.14  0.95 g / cc
 (9.807m / s 2 ) 
 
 1000 g   m 


 6.20 x10  3 m
2

 3 kg 
2



(3000cP)  10


m
s
cP


 115
.
3


 kg   100cm 
2
(0.95g / cc) 




 1000 g   m 


 ut 


kg 
 (3000cP) 10  3
 (7.14  0.95g / cc)(9.807m / s 2 ) 
m s cP 





 ut 170
. s/m
From the figure ut*  0.07 hence ut 
1/ 3
0.07
 0.041m / s .
170
. s/m
Similarly for the water
ut*  ut  2551
. s / m
d p*  243
and from the figure ut*  17 hence ut  0.67m / s .
This shows that a change of 3000x in viscosity produces about a 10x change in
the terminal velocity.
3-29
1/ 3
SOLIDS NOTES 3, George G. Chase, The University of Akron
More on Sphericity:
We represent a bed of non-spherical particles by a bed of spheres of diameter Deff such
that a bed of spheres and a bed of non-spheres have

The same total surface area, a in a given volume of the bed.

The same fractional voidage, bed .
This representation ensures almost the same flow resistance in both beds.
In typical use of the Ergun Equation (McCabe & Smith), the effective diameter of the
particle is replaced with the sphericity times the defined diameter based on sphericity;
Deff  Dsph .
The sphecific surface area of particles in either bed is found to be
as
 Surface area of one particle 


 volme of one particle 
 D 2sph / 
=
3
 Dsph
/6
6

Dsph
(3-69)
For the whole bed

 6(1   )
Surface of all particles
a

Dsph
 Total volume of particles in the bed 
(3-70)
Since there is no general relationship between Deff and d p (particle diameter
corresponding to a sphere of the same volume), the best we can do without running
experiments is as follows:

For irregular particles with no seemingly longer or shorter dimensions (hence
isotropic in irregular shape)
Deff  Dsph  d p

For irregular particles with one longer direction , but with a length ratio not
greater than 2:1 (eggs for example)
Deff  Dsph  d p

(3-72)
For irregular particles with one dimension shorter, but with a length ratio not
less than 1:2 (peanut, for example)
Deff  Dsph   2 d p

(3-71)
(3-73)
For very flat or needlelike particles, estimate the relationship between d p and
Deff from  values for corresponding disks and cylinders.
3-30