SolidsHandouts3.pdf

7/30/2019 SolidsHandouts3.pdf

1/12

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


2/12

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


3/12

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.


4/12

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


5/12

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.


6/12

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.


7/12

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


8/12

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


9/12

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


10/12

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


11/12

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.


12/12

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

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