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7/30/2019 SolidsHandouts3.pdf
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HANDOUT 3.1
1 Angstrom
1 Nanometer
1 Micron
1 Millimeter
1
10
100
10001
10
100
10001
10
100 APPROX.MOLEC.
WT.
100
200
20k
200k
AQUEOUSSALTS
CARBONBLACK
PAINTPIGMENT
BACTERIA
YEAST CELLS
PROTEIALBUMI
TALCCLAY
RED BLOCELLS
POLLEN
HUMAN HAIR
RANGEPARTICLE
MATERIALSCOMMON
LOG SCALE
PARTICLE SIZE
MOLECULA
R
Ultraviolet
X-rays
SPECTRUMMAGNETICELECTRO-
0
1
2
3
ION
IC
MOLECULE
MACR
O
Infrared
Radio waves
7
4
5
6
8
MAC
RO
Visible
MICRO
COLLOIDALMICROSCOPE
ATOMS
METAL IONS
SUGARS
VIRUS
SILICA
ELEC
TRON
MICROSCOPE
PYROGEN
TOBACCO SMOKE
BEACHSAND
GRAVEL
VISIBLETO
EY
E
OPTICAL
DUST
MILLEDFLOUR
COAL
-LUCITE-GEON-ETC.
POLYMERPOWDERS
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HANDOUT 3.2
STANDARD MESH SIZE
Tyler US mm Inches
4 4 4.70 0.185
6 6 3.33 0.131
8 8 2.36 0.094
10 12 1.65 0.065
12 14 1.40 0.056
14 16 1.17 0.047
16 18 0.991 0.039
20 20 0.833 0.033
24 25 0.701 0.028
28 30 0.589 0.023
32 35 0.495 0.020
35 40 0.417 0.016
42 45 0.351 0.014
48 50 0.295 0.012
60 60 0.246 0.0097
80 80 0.175 0.0069
100 100 0.147 0.0058
150 140 0.104 0.0041
200 200 0.074 0.0029
250 230 0.061 0.0024
325 325 0.043 0.0017
400 400 0.038 0.0015
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HANDOUT 3.3
Taken from Tables 2.1, 2.2, 2.3, and 2.7 in L. Svarovsky, Solid-Liquid Separation, 3rdEd., Butterworths,
London, 1990.
DEFINITIONS OF EQUIVALENT AND STATISTICAL DIAMETERS.
Symbol Name DefinitionDEFINITIONS OF EQUIVALENT SPHERE DIAMETERS
xv Volume diameter Diameter of sphere with the same volume as the particle.
xs Surface diameter Diameter of sphere with the same surface area as the particle.
xd Drag diameter Diameter of sphere that has the same resistance to motions at the
same velocity as the particle.
xf Free-falling diameter Diameter of sphere of same density as the particle with the same
free-falling speed in the same liquid.
xSt Stokes diameter Same as xfbut for when Stokes Law applies (Re < 0.2)
xA Sieve diameter Largest diameter sphere that can pass through the square aperture
of the sieve screen.
xSV Surface to Volume Ratio Diameter of sphere that has the same surface area to volume ratio
as the particle.
DEFINITIONS OF EQUIVALENT CIRCLE DIAMETERSxz Projected area diameter Projected area if the particle is resting in a stable position.
xp Projected area diameter Projected area if the particle is randomly oriented.
xc Perimeter diameter Diameter of a sphere with the same projected perimeter as the
perimeter of the projected outline of the particle.
DEFINITIONS OF STATISTICAL DIAMETERS
xF Ferets diameter Distance between two tangents on opposite sides of the particle.
xM Martins diameter Length of the line which bisects the projected image of the particle
(the two halves of the image have equal areas).
xSH Shear diameter Particle width obtained with an image shearing eyepiece.
xCH Maximum chord
diameter
Maximum length of a line limited by the contour of the projected
image of the particle.
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HANDOUT 3.4
LABORATORY METHODS OF PARTICLE SIZE MEASUREMENTS
METHOD APPROX
SIZE, mSIZE TYPE TYPE OF SIZE
DISTRIBUTION
Sieving (wet or dry)
Woven wireElectro formed
37-40005-120
xA By mass
Microscopy
Optical
Electron
0.8 150
0.001 5
xz, xF, xMxSH, xCH
By number
Gravity sedimentation 2-100 xSt, xf By mass
Centrifugal sedimentation 0.01 - 10 xSt, xf By mass
Flow Classification
Gravity elutriation (dry)
Centrifugal elutriation (dry)
Impactors (dry)
Cyclonic (wet or dry)
5 - 100
2 - 50
0.3 50
5 - 50
xSt, xf
By mass
By mass
By mass or by number
By mass
Coulter principle (elect. resist.) 0.8 200 xv By number
Field flow fractionation 0.001 100 xd Depends upon detectorHydrodynamic chromatography 0.01 50 xd Depends upon detector
Fraunhofer diffraction (laser) 1 2000 Equiv laser diameter By volume
Mie theory light scattering (laser) 0.1 40 Equiv laser diameter By volume
Photon correlations spectroscopy 0.003 3 Equiv laser diameter By number
Scanning infrared laser 3 100 Chord length By number
Aerodynamic sizing nozzle flow 0.5 30 xd By number
Mesh obscurtion method 5 25 xA By number
Laser Doppler phase shift 1 10,000 Equiv laser diameter Mean only
Time of transition 150 1200 Equiv laser diameter By number
Surface area to volume ratio
Permeametry
Hindered settling
Gas diffusion
Gas adsorption
Adsorption from solution
Flow microcalorimetry
Calculated xSV By number mean
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HANDOUT 3.5
ELECTRONIC PARTICLE COUNTER
The electronic particle counters can measure particle sizes ranging from 0.4 to 1200 micrometers. This
method requires the particles to be placed in a stirred electrolyte solution. The resistance to the flow ofelectrical current through a small aperture is calibrated to the change in resistance depending upon the
particle size (Figure 1).
Figure 1. Basic components of the Coulter Counter.
As the particles pass through the aperture opening, they bend the current flux lines around the particles,
thus causing a longer length for the current to pass and thus a higher resistance to the current (Figure 2).Voltage and current are measured to quantify the resistance using Ohms Law: V = IR.
APERTURE OPENING APERTURE OPENING
WITHOUT PARTICLE WITH PARTICLE
Figure 2. Particles in the aperture bend the electrical current flux lines.
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HANDOUT 3.6EXAMPLE 3-1
A sample of M&Ms 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.
Table 3-1. Mass and diameter distribution of M&Ms.
Grams Dia, cm Size < Avg size No. fdx f F
2.06 1.473 1.5 1.475 1 0.047619 0.952381 0.047619
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 0.333333 6.666667 0.3809522.4 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 0.428571 8.571429 0.809524
2.71 1.614 1.65 1.625 1 0.047619 0.952381 0.857143
2.94 1.659
2.99 1.668 1.7 1.675 2 0.095238 1.904762 0.9523813.01 1.672 1.75 1.725 1 0.047619 0.952381 1
Using the formulas in
Eqs.(3-13) and (3-14) the
frequency and cumulative
frequency distributions are
calculated. The particle
sizes are added up in
increments of 0.05 cm. The
size ranges start with 1.45 to
1.50 cm. All M&Ms of size
less than 1.50 are counted in
the first increment, all
M&Ms with size between
1.5 and 1.55 are in the
second increment, and so on.
x
The values for nj are
determined by counting the
number of M&Ms that fall
in a given size increment
and are assigned to theaverage size in the
increment.
For example, there are 7
M&Ms in the size increment
range of 1.5 to 1.55 cm and
are assigned to the average
size of 1.525 cm.
Frequency Distribution of M&Ms
0
2
4
6
8
10
1.45 1.5 1.55 1.6 1.65 1.7 1.75
Diameter, cm
FrequencyDistribution
0
0.2
0.4
0.6
0.8
1
f
F
Figure 3-4. Plot of frequency and cumulative frequency distributions for
M&Ms.
fdx is determined by
7/21=0.33333, f is0.33333/0.05 = 6.66667. F
is determined by cumulative
summing the values fdx.
The results of the
summation are plotted in
Figure 3-4.
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HANDOUT 3.7MODE
HARMONIC MEAN
ARITHMETIC MEAN
MEDIAN
f
QUADRATIC MEAN
CUBIC MEAN
f
x
Figure 3.5. Comparison of mean size distributions
where the various means are defined by:
( )g x g x dF = ( )01
g(x) = NAME OF MEAN
xARITHMETIC MEAN, ax
x2 QUADRATIC MEAN, qx
x3 CUBIC MEAN, cx
log xGEOMETRIC MEAN, gx
1/x HARMONIC MEAN, hx
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HANDOUT 3.8
Sieve analysis of a sample of particles. Mass, number, and area fractions are calculated.
Sieve analysis of a sample of particles. Mass, number andarea fractions are calculated.
Note 1 Note 2
SIEVE AVG SIEVE MASSVOLUMEON VOLUME V1 NUMBER NUMBER A1
AREATRAY AREA
SIZE,MM SIZE, MM
MASS,g FRAC
TRAY,MM 3 FRAC MM 3 FRAC MM 2 MM 2 FRAC
pan 0
0.04 0.05 0.10 0.03 38.46 0.03 0.00 67293.01 0.44 0.01 518.00 0.11
0.06 0.08 0.40 0.11 153.85 0.11 0.00 58141.16 0.38 0.02 1243.20 0.25
0.10 0.14 0.70 0.19 269.23 0.19 0.01 21045.58 0.14 0.06 1286.65 0.26
0.18 0.24 0.90 0.25 346.15 0.25 0.06 5660.10 0.04 0.17 982.00 0.20
0.30 0.36 0.70 0.19 269.23 0.19 0.21 1266.29 0.01 0.40 504.18 0.10
0.42 0.50 0.50 0.14 192.31 0.14 0.60 320.67 0.00 0.79 254.88 0.05
0.59 0.71 0.20 0.06 76.92 0.06 1.69 45.42 0.00 1.59 72.13 0.01
0.83 0.92 0.10 0.03 38.46 0.03 3.63 10.60 0.00 2.64 27.98 0.011.00
TOTAL MASS 3.60 1.00 1384.62 1.00 153782.82 1.00 4889.01 1.00
Comparison of the fractional distributions of the particle size distributions.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.00 0.20 0.40 0.60 0.80 1.00
Avg Part ic le Size, mm
Fraction Mass & Volume Frac
Number Frac
Area Frac
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HANDOUT 3.9
0.1
1
10
100
1000
10000
100000
0.001 0.01 0.1 1 10 100 1000 10000 100000
Re
Cd
Expl curve
Stokes
Intermediate
Newton Law
Figure 3.9. Drag coefficient for spheres versus Reynolds number. The three approximate curves from left
to right are (Stokes Law range for RC RD e= 24 / p Repep
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HANDOUT 3.10
Table 3-3 Sphericity of Some Common Materials (McCabe & Smith, 5th ed, pg928; Perrys Handbook 6th
ed, pg 5-54).
PARTICLE MATERIAL SPHERICITY
Sphere 1.0
Cube 0.81
Short Cylinder (Length=Diameter) 0.87
Berl saddles 0.3
Raschig rings 0.3
Coal dust, natural (up to 3/8 inch) 0.65
Glass, crushed 0.65
Mica flakes 0.28
Sand
Average for various types
Flint sand, jagged
Sand, rounded
Wilcox sand, jagged
0.75
0.65
0.83
0.6
Most crushed materials 0.6 to 0.8
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HANDOUT 3.11
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
1.E+10
1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04
Rep
CdRep^2
SPHERICITY
0.8
0.6
1.0
0.4
0.2
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.
Where GAepd NRC 342 =
udR
p
ep = ( )
2
3
gdN
pp
GA
=
pD is the equivalent diameter of a sphere with the same volume as the particles, xv.
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HANDOUT 3.12
0.01
0.1
1
10
100
1 10 100 1000 10000
dp*
ut*
Sphericity = 1.0
0.9
. . .
0.5
0.23
0.123
0.043
0.026
Sphericity for
Disks only
Date taken from
Kunii & Levenspiel
Fluidization Engineering, 2ed
Butterworth, Boston, 1991, page 81
Where
( )
3/12
*
=
guu
gs
g
tt
and
( ) 3/12
*
=
gdd
gsg
pp