Dispersity of particulate systems, About rocks, gravel ...€¦ · 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 1 1 o A 1 nm 1 µm 1 mm 1 cm 1 m wave length of visible light:

Fig. 1.1

Prof. Dr. J. Tomas, chair of Mechanical Process Engineering

Fig_MPE_1 Mechanical Process Engineering - Particle Technology Disperse Systems Prof. Dr. J. Tomas 31.03.2014 Figure 1.1

1. Dispersity of particulate systems,

1.1 About rocks, gravel, lumps, nuggets, corn, particles, nanoparticles and colloids

1.2 Particle characterisation - Granulometry, 1.3 Particle size distributions, 1.4 Physical particle properties

Fig. 1.2



Size Scale of Polydisperse (Material) Particle Systems

10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 1

1

o

A 1 nm 1 µm 1 mm 1 cm 1 m

wave length of visible light: visual ability of human eye

X-rays and electron interferences

ultra-microscope light microscope

electron microscope

capacitive und inductive sensors

dispersity molecular-disperse colloid-

disperse high-disperse,

ultra-fine fine-disperse coarse-disperse

pore dispersity microporous mesoporous macroporous dispersed elements

molecules makromolecules, colloids

ultra-fines fines medium grain coarse

one-dimensional surface coatings, liquid films, membranes two-dimensional chains of macromolecules, needles, fibres, threads

Fig. 1.3



Blatt 2 Mixtures of Polydisperse (Material) Particle Systems

10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 1 disper-sant

disperse phase

1 o

A 1 nm 1 µm 1 mm 1 cm 1 m

gas gas gas mixture liquid aerosol, fog solid aerosol, smoke

transition l-g foam liquid gas solution, lyosol,

hydrosol bubble system

liquid micro-emulsion emulsion solid suspension

solid gas xerogel, porous membrane rigid-foam insulation liquid gel liquid filled, porous solid material solid mixed crystal, solid solution, s-s alloy

monodisperse = uniform-sized elements

Fig. 1.4



1 10 100 1000 nm

10–9 10–6 m

Quantum effects

Strongly developed surface effects

Polymers

Ceramic powders

Tobacco smoke

Nanoparticles for life sciences

Bioavailability

Proteins

Virus, DNS

Atmospheric aerosols

Metal powders

0,001 0,01 0,1 1 µm

Size Scale and Properties of Nanoparticles

Fig. 1.5



Expression of the Particle Size characte-ristic size

Eq./sketch measuring method, quantity r = 0...3

breadth: b length: l thickness: t 2/1

3/1

6lt2bt2lb2,lb,

b/1l/13

,tb,

3tlb,

21b

++

+

+++

(1) equivalent diameter d, for b ≈ l ≈ t (2) equivalent length l, for rods l >> b ≈ t (3) equivalent area lb ⋅ , for chips, plates b ≈ l >> t (4) equivalent mass tlbs ⋅⋅⋅ρ , for extreme

shaped clusters:

r = 0 number basis r = 1 length r = 2 area r = 3 volume basis image analysis d0 geometric anal. d0 geometric anal. d0 mass balancing d3

Feret diameter

image analysis, number basis d0

Martin diameter

21 AAA +=

image analysis, number basis d0

sieve diameter

( ) 2121 aaoraa21

+

sieving, mass or volume basis d3

volume V equivalent diameter

equivalent volume diameter 3 /V6 π⋅

Coulter counter electrical method, number basis d0

area A equivalent diameter

equivalent projection area diameter π/A4

light extinction, number basis d0

surface area AS equiv. diameter

equivalent surface area diameter π/AS specific surface diameter SA/V

light extinction, number basis d0

physical feature equivalent diameter

Stokes diameter

( ) a18v

dfs

sSt ⋅ρ−ρ

η⋅⋅=

gravitational, centri-fugal sedimentation and impactor, mass basis d3

aerodynamic diameter a18v

d sa

η⋅⋅= sedimentation, mass

or volume basis d3 equivalent light-scattering diameter

low angle laser light-scattering method, number basis d0

b

t

a1 a2

vs

Fig. 1.6



Characterisation of particle size distributions

1. Particle size characteristics by 2. Particle size distribution function image analysis (cumulative distribution curve)

3. Frequency distribution of particle size (distribution density curve)

5. Particle size distribution function Q3(d) and frequency distribution of particle size q3(d) of the above example 4.

4. Example of measured particle size distribution

a) b)

dF FERET chord lengthdM MARTIN chord lengthdS maximum chord length

particle sizefractiondi-1 ... diin mm

mass

in kg

massfraction

Q3(di)-Q3(di-1)in %

cumulativefraction

Q3(d) in%

- 0.16 0.16 ... 0.63 0.63 ... 1.25 1.25 ... 2.5 2.5 ... 5.0 5.0 ... 6.3 6.3 ... 1010 ... 1616 ... 20 + 20

0.1800.6480.9191.9203.0211.0841.7480.7610.2320.054

1.7 6.1 8.718.128.610.316.6 7.2 2.2 0.5

1.7 7.8 16.5 34.6 63.2 73.5 90.1 97.3 99.5100.0

10.567 100.0

dire

ctio

n of

mea

sure

men

t

dFdMdS

0.20

0.15

0.1

0.05

00 4 8 12 16 20

particle size d in mm

dm,i =di-1 + di 2

qr(d) ≈ Qr(di) - Qr(di-1) di - di-1

q 3(d

) in

mm

-1

1

0,5

0 dmin d1 d2 dmaxd

∆Qr(d)Qr(d2)

Qr(d1)

Qr(d)

∆d

qr(d)

du di-1 di di+1 d0d

qr(d) = dQr(d)d(d) Mode dh

0 4 8 12 16 20Particle size d in mm

100 80

60

40

20

0

Q3(d

) in

%

Qr(d*<di)

Median d50

Characterisation of Granulometric Properties of Disperse Material Systems

Fig. 1.7



Normal Distribution (GAUSSIAN Distribution):

Four - Parameter Log - Normal Distribution:

WEIBULL Distribution:

0 5 10 15 x

qr(x)0.3

0.2

0.1

ln x50 = 1, σln = 1

ln x50 = 3, σln = √3

ln x50 = 3, σln = 1

qr(x)

2.0

1.0

n = 0.5 n = 5.5

n = 3

n = 2

n = 1

for xmin = 0 and x* = x63 = 1

0 1 2 x

( )

( )

( )

( )2

xx4xxu

with

dt2texp

21xQ

:normalizes

dtt21exp

21xQ

x21exp

21xq

168450

u 2

r

x 2

r

2

r

−=σ

σ−

=

−

π=

σµ−

⋅−π⋅σ

=

σµ−

⋅−π⋅⋅σ

=

∫

∫

∞−

∞−

( )

( )

( )

⋅=σ

σ−

=

≤≤⋅−

−=

σ−

−⋅π⋅σ

=

σ−

⋅−⋅π⋅⋅⋅σ

=

∫

16

84ln

ln

50

maxminmaxmax

min

x

0

2

ln

50

lnr

2

ln

50

lnr

xxln

219xlnxlnu

dddforddd

ddx

dtxlntln21exp

t1

21xQ

xlnxln21exp

2x1xq

( )

( )

−−

−−=

−−

−

−−

−=

∗

∗

−

∗∗

n

min

minr

n

min

min

1n

min

min

minr

xxxxexp1xQ

xxxxexp

xxxx

xxnxq

(1)

(2)

(3)

(5)

(6)

(7)

(8)

(10)

(11)

(12)

(13)for xmin = 0 and n = 1 follows theExponential Distribution if λ =

Q(x) = 1 - exp(-λ·x)

1x63

Typical frequency distributions and cumulative probability distributions

σ2 < σ1

σ1

qr(x)

0 x16 xh = x50 x84 x

Qr(x50) = 0,5

σ σ

Mode xh = x50 Median

Fig. 1.8



6. Three - parameter logarithmic normal distribution (L) with upper limit do and transformation (T)

7. Comparison of particle size distribution functions in a full-logarithmic, RRSB and log - normal diagram (net)

1 5 10 5 10 5

99.9099.509790

50

1051

0.200.02

Qr(d

) L

d50 do

δ16 δ50 δ84d or δ

T

3 - parameterdistribution

transformeddistribution

8. RRSB - distribution in a double - logarithmic diagram

AS,V,K · d63 in m3/ m3

n 40

60 80 100 120 150 200 300 500 1000 2000 500010000

10-3 10-2 10-1 100 101 102

99.9999590

63.250

10

1

0.5


Pol

cum

ulat

ive

dist

ribut

ion

Q3(d

) in

% xx

xxx

x

x

x

x

0

0.1

0.3

0.2

0.4

0.5

0.6

0.7

0.8

0.9

1.01.11.21.31.41.61.82.02.54.0 3.03.5

10 15 20 25 30987.57.0

1 Log-Normal distribution2 RRSB-distribution3 GGS-distribution

particle size d in µm

100 101 102 103 104

99.96040

20

6

0.5

10cu

mul

ativ

e di

strib

utio

n Q

(d) i

n %

50

5

full-

loga

rithm

ic n

et

RR

SB -

net

2

4

10

11

0.5

510

99.999.5

98969080604020

1 2 3 1 2 3 1 2 3

full-logarith-mic -net

RRSB-netlog - normalnet

99,995

10-1 100 101 102 103 10-2 10-1 100 101 102

Graphical characterisation of selected particle size distributions

Fig. 1.9



Statistical Moments of Particle Size Distributions

Complete k-th Moment of Particle Size Distribution Qr(d*<d) related to Quantity r:

( ) ( ) ( )d k rk

rd

dk

rd

d

m r i

k

r ii

N

M d d q d d d d d dQ d d du

o

u

o

* ,* *

, , ,( ) ( ) ( )= − ⋅ = − ⋅ ≈ − ⋅∫ ∫ ∑ ∗

=

µ1

(1)

First Initial Moment (k = 1, d* = 0) or Expected Value

M d d q d d d d dQ d dr m r rd

d

rd

d

m r i r ii

N

u

o

u

o

11

, , , , ,( ) ( ) ( )= = ⋅ = ⋅ ≈ ⋅∫ ∫ ∑=

µ (2)

Central Moment related to expectation (mean) dm,r

d k r k r m rk

rd

d

m r

u

o

M Z d d q d d d, , , ,( ) ( ) ( )= = −∫ (3)

Second Central Moment or Variance

Z d d q d d d d d dQ d d dr r m r r m r r m r i m ri

N

d

d

d

d

r iu

o

u

o

22 2 2 2

1, , , , , , ,( ) ( ) ( ) ( ) ( ) ( )= = − = − ≈ − ⋅

=∑∫∫σ µ (4)

Variance according to “Satz von Steiner”

σ µr r r r m r i r i m ri

N

Z M M d d22 2 1

2 2 2

1= = − ≈ ⋅ −

=∑, , ,, , , , , (5)

Incomplete k-th Initial Moment du...d, i...n and Complete Initial Moment du...do, i...n...N

d q d d d dk

u

m r ird

dk

i

n

r i( ) ( ), , ,∫ ∑≈ ⋅

= 1µ (6) d q d d d d

k

u

o

m r ird

dk

i

N

r i( ) ( ), , ,∫ ∑≈ ⋅

= 1µ (7)

Conversion from given quantity r to a searched quantity t of Frequency Distribution

q dd q d

Mt

t rr

t r r( )

( )

,=

⋅−

−

(8)

and Cumulative Distribution

Q dM

M

d q d d d

d q d d d

d

dt

t r r d

d

t r r d

d

t rr

d

d

t rr

d

d

m r it r

r ii

n

m r it r

r ii

Nu

u

o

u

u

o( )

( ) ( )

( ) ( )

,

,

, , ,

, , ,

= = ≈⋅

⋅

−

−

−

−

−

=

−

=

∫

∫

∑

∑

µ

µ

1

1

(9)

Conversion of cumulative distributions from number to mass basis or from mass to number basis

Q dd q d d d

d q d d d

d

d

d

d

d

d

m i ii

n

m i ii

Nu

u

o3

30

30

03

01

03

01

( )( ) ( )

( ) ( )

, , ,

, , ,

= ≈⋅

⋅

∫

∫

∑

∑=

=

µ

µ (10) Q d

d q d d d

d q d d d

d

d

d

d

d

d

m i ii

n

m i ii

Nu

u

o0

33

33

33

31

33

31

( )( ) ( )

( ) ( )

, , ,

, , ,

= ≈⋅

⋅

−

−

−

=

−

=

∫

∫

∑

∑

µ

µ (11)

Conversion of k-th complete initial moment of a known quantity r in a searched quantity t

MMMk t

k t r r

t r r,

,

,= + −

−

(12)

Fig. 1.10



Cumulative Particle Size Distribution, Mass and Number Basis

mass basis: ∫ ∑=

µ≈⋅=d

d

n

1ii,333

U

)d(d)d(q)d(Q

number basis: ∑

∑

∫

∫

=

=

−

−

µ

µ

≈

⋅⋅

⋅⋅

=N

1i3

i,m

i,3

n

1i3

i,m

i,3

d

d3

3

d

d3

3

0

d

d

)d(d)d(qd

)d(d)d(qd)d(Q

o

u

u

0

10

20

30

40

50

60

70

80

90

100

Verteilungsfunktion Q0(d) in %

0.5 1 5 10 50 100 500 1000

Partikelgröße in µm

Anzahlverteilung

0

10

20

30

40

50

60

70

80

90

100

Verteilungsfunktion Q3(d) in %

0.5 1 5 10 50 100 500 1000Partikelgröße in µm

Masseverteilung

Fig. 1.11



Multi-modal Frequency Distribution

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8


freq

uenc

y di

stri

butio

n q*

(log

d)

subcollective 3

subcollective 2

subcollective 1

0.1 100.010.01.0

total frequency distribution:

[ ]q d t q d d dtot SC k k o k k kk

N

3 3 501

, , , , , ln,( , ) , , ,= ⋅∑=

µ σ

truncated log-normal distribution:

q dd d

d du

ko k

k o k3

2

2 2,,

ln, ,( ) exp=

−

⋅ ⋅ ⋅⋅ −

π σ

with

ud dd d

d dd dk

o k

o k

o k k

o k k=

⋅

−

−

⋅

−

1 50

50σ ln,

,

,

, ,

, ,ln ln

normalisation:

( )( )

( )q

dQ dd d

Q dd d

d

toti

i

i

33 3 3

1

,* ,log

loglog

loglog

= ≈ =

−

∆∆

µ

µSC,k(t) mass fraction of the k-th

subcollective (subpopulation)

q3,k frequency distribution

of the k-th subcollective

do,k upper limit of the particle size

of the k-th subcollective

d50,k median particle size of the

distribution function

σln,k standard deviation of the

k-th subcollective

N total number of subcollectives

[ ] )t,d(Q)d(d2

uexp21,d,d,dQlim tot,3

u 2

1kkln,k,50k,ok,3k,SCN

=

−

π=σ⋅µ ∫∑

∞−

∞

=∞→

Fig. 1.12



Mass Fraction Related to the Number of Stressing Events

discrete mass balance model:

1,sc1,n1,sc S

nd

µ⋅−=µ

2,sc2,3,n1,sc1,3,n3,sc SS

nd

µ⋅+µ⋅=µ

1

N

1kk,sc =µ∑

=

n number of stressing events Sk,j kinetic constants for mass transfer from j to k subcollective

0 1 2 3 43

2

1

0,00,20,40,60,81,0

3

number of stressing events n

mass fraction µsc,k

k-th subcollective

measuredmodel

Fig. 1.13



Application of Image Analysis to Characterise Particle Size 1. Image of Microscope by CCD-camera

2. Definition of Threshold Value 3. Conversion of Grey Tone Image in a Binary Image (Binarisation) 4. Classification of Particles

dF,mi

dF,ma

Definition of grey tone limits for particle detection in a 8-bit grey tone image

Binary image means: which pixel of original image is shown by 0 (black) or by 255 (white)

dequ

• min. and maximum Feret diameter • equivalent circle diameter

π/Ad ⋅= 2 ,

• shape factor 24UA

U ⋅⋅= πψ

U = circumference, A = projection area

Presentation of Particle Size Fractions in a Colour Code

direct-light

transmitted light

pixe

l num

ber

grey tone distribution0 255(black) (wite)

particle

direct-light

transmitted light

Fig. 1.14



Principle of Laser Light Diffraction large diffraction for particle size d ≈ λ wavelength, small diffraction for d >> λ

light diffraction pattern radial light intensity

distribution at detector

∫ ⋅⋅=max

min

d

di0tottot )d(d)d,r(I)d(qNI

particle size distribution

laser

lense systemsample cell Fourier

lensedetector

r

computer

ffocal distance

principle of laser light diffractometer

r

Fourier lense detector

prinziple of Fourier lense

Intensity I

r

100

50

0particle size


cum

ulat

ive

dist

ribu

tion

Q3 i

n %

freq

uenc

y di

stri

butio

n q 3

in 1

/mm

Fig. 1.15



In-Line Particle Size Analysis (Sympatec)

isokinetic sampling device for a split particle stream: rotating sector moving pipe

D

α

D

d

particle loaded air stream

drive for rotating sampling device

dispersion air

laser beam

detector with sensor array

nozzle and sample cell

low-angle laser light-scattering instrument (LALLS) d = 0.5 – 1750 µm

inductive sensor

on-line sampling

feed opening

Fig. 1.16



In-Line Particle Size Analysis (Malvern)

monitoring of size ranges: 0.5 - 200 µm 1.0 - 400 µm 2.25 - 850 µm

Injektion nozzle

laser

pressurized air

particlestream

isokinetic sampling

particle feed back

detector

Fig. 1.17



Principle of Photon – Correlation Spectrometer (PCS) in suspensions at rest: light scattering at dispersed particles, that oscillate by Brownian

molecular motion

Determination of intensity – time function of scattered light (reasons: interferences,

change of particle number concentration within the charac-teristic volume element) and calculation of autocorrelation function:

• Autocorrelation function (Dp – particle diffusion coefficient, K – scattered light vector,

τ - retardation time)

τ⋅⋅⋅−

−∞→

=τ+⋅=τ ∫2

p KD2T

TTI,I edt)t(I)t(Ilim)(R

with p

B

D3Tkd⋅η⋅π⋅

⋅=

• EINSTEIN equation (d – particle size, kB – BOLTZMANN constant, T – absolute temperature, η - dynamic viscosity)

auto

corre

latio

n fu

nctio

n R

I,I( τ

)

retardation time τ

inte

nsity

of s

catte

red

light

time t

fine particle

coarse particle

Laser Optik Probenbehälter

Photomultiplier KorrelatorOptische Einheit

Fig. 1.18



Laser

Detectorsbackscatter

large angleforward angle

Fourier lens Sample chamber

Laser




Laser




Laser




1. Physical Principle

Laser diffraction technique is based on the phenominon that particles scatter lightin all directions (backscattering and diffraction) with an intensity that is dependenton particle size

- the angle of the deflected laser beam is inverse proportional to the particle size

2. Measurement setup

Using two laser beams with different wavelength (red and blue light) additional information to particles smaller 0,2 µm is obtained

red light setup

- scattering light hits only forward angle detectors

blue light setup

- blue light (wavelength 466 nm) leads to a scattering signal for small particles (isotropic scattering pattern) which can be detected from large angle- and backscatter- detectors

Θ

Θ

page 1

Fig. 1.19



Fig. 1.20



Principle of Acoustic Attenuation Spectroscopy

during acoustic wave penetration, amplitude and intensity attenuation (damping) of

ultrasonic frequency spectrum (1 to 100 MHz) in high concentrated particle suspensions with sizes d = 10 nm – 1 mm

detection of attenuation (damping) spectrum correlation between attenuation characteristics

and particle size distribution (K = 2⋅π/λ suspension wave number, k fluid wave number, ϕs particle volume concentration, i = 1...n particle size fraction, ri particle radius, Ami reflected compression wave coefficient, ARe real contribution, m number of acoustic dispersion coefficient):

( ) miRe0m

n

1i3

i3

i,s2

AA1m2rk

i231

kK

⋅+⋅

ϕ⋅−=

∑∑

∞

==

Microwave and

DSP module

TransducerPositioning Table

Controlmodule

Discharge

Stopper motorand digitalencoder

Level sensor

Suspension

HF Receiver

LF Receiver

HF Transmitter

LF Transmitter

Stirrer

entrainmentx << λ

x >>scattering

λλ

RF generator RF detector

measuring zone

100

50

0particle size


cum

ulat

ive

dist

ribu

tion

Q3 i

n %

freq

uenc

y di

stri

butio

n q 3

in 1

/mm

dam

ping

frequency

Fig. 1.21



Determination of Particle Size Distribution and Zeta-Potential usingElectroacoustic Effect - Electrokinetic Sonic Amplitude (ESA)

1. Physical PrincipleAlternating electric field (frequency range 1 to 20 MHz) generates particle oscillationsat velocities that depend on their size and zeta potential (O' Brien- Theory)

2. Measurement Setup

3. Data Analysis

adjusting q(d) and zeta-potential ζ fromthe measured mobility spectrum

Ep

s ZAESA µρ

ρϕω ⋅⋅∆

⋅⋅= )(

A(ω) calibration function ϕs volume fraction of particles ∆ρ suspension density difference ρp particle density Z acoustic impedance (complex resistance)

( )∫ ⋅= )()(,, dddqd sEm ϕζµµ

µm measured dynamic mobility ζ zeta-potential d particle diameter ϕs volume fraction of particles q(d) particle size frequency distribution

acoustic signal (ESA) as response

∆ρ ∼ ∆p

ηζεεµ ⋅⋅== rE E

v0

electrophoretic mobility (µE)

suspension

ESA-SignalProcessing

Particle motion in an electric field

Time

E;v

applied electric field particle velocity

ε0 permittivity of vacuum εr permittivity v particle velocity E electric field strength η viscosity

frequency

phas

e la

g

Mobility Spectrum

µm

dyn. mobility

phase lag

Fig. 1.22



Particle Density Measurement by HELIUM-Pycnometer Determination of pore–free particle volume by gas pressure measurement in a double-

chamber system by HELIUM gas (migration access of internal pores dPore > 0,1 nm)

• Pressure measurement in probe chamber: (VCell –VProbe) p1 • Pressure test in probe and expansion chamber: (VCell –VProbe) + VExp p2

• Calculation of probe volume and solid density, pre-measurement of particle mass ms by balance

1p/pV

VV21

ExpCellobePr −

−= and obePr

ss V

m=ρ

pressureprobe chamber

filterHelium

feed valve

overpressurevalve

prep./ test valve

discharge valve

VProbe

V Ex

p 5

V Ex

p 35

V Ex

p 15

0

VCell 5,

VCell 35,VCell 150

P

Fig. 1.23



Measurement of Particle Surface by Gas Adsorption according to BRUNAUER, EMMET and TELLER

Physical adsorption of gas molecules at particle surfaces in multi-layers due to VAN DER WAALS interaction

BET- line, valid for: 0.05 < p/p0 < 0.3 • Adsorpt mono-layer coverage:

ba

1V mono,g +=

• BET- constant:

aba

TRHH

expC multimBET

+=

⋅∆−∆

=

∆H m free molar adsorption enthalpy of mono-layer

∆H multi molar bonding enthalpy of n multi-layers ≅ ∆Hcondensation

• Particle Surface:

l,mmono,gAg,MS V/VNAA ⋅⋅= AM,g cross-sectional area of adsorpt

molecule NA AVOGADRO-number Vm,l molar volume of condensed adsorpt

( ) 0BETmono,g

BET

BETmono,g0g

0 p/pCV

1CCV

1p/p1V

p/p⋅

⋅−

+⋅

=−

gas supplyP

PTdosingvalve

probe chamber

dewar vesselp0 - test chamber

liquid nitrogenN2 at T = 77 Kp0 = 101 kPa

T

vacuum

standard vessel

0 0.35 1

adso

rbed

gas

v ol

ume

Vg

desorption

adsorption

BET range sorption isotherms

relative partial pressure of gas p/p0

adsorbedgas molecules(adsorpt)

adsorptiv

particle surface(adsorbens)

( )0g

0

p/p1Vp/p

−

0p/p

BETmono,g CV1a⋅

=

( )BETmo n o,g

BET

CV1Cb

⋅−

=

relative gas pressure

Fig. 1.24



Regular Packing Structures

porosity ε, coordination number k

lattice type primitive basic face- face-centred space-centred centred

β α

γ

z c

by

x a

cubica = b = cα = β = γ = 90 °

monodispersesphere packingd = const.

hexagonala = b = cα = β = 90 ° γ = 120 °

sphere packing

a0 0,1nm k = 6 k = 12 k = 8≈

ε = 0,4764 ε = 0,3955

k = 12

ε = 0,2595

a0

d

octahedronvacancy

tetrahedronvacancy

Fig. 1.25



Fig. 1.26



Stressing and Flow of Wet Particle Dispersions

ad > 1 0 < < 0.2

ad

ad = 0

ϕss< 0.066 0.3 < ϕs <

π6

εs,0 =π6

pore saturation S = 1

ϕi = 0 ϕi ≥ 0ϕi = 0

particle in liquid dispersion (suspension) paste liquid in particle packingdiluted concentrated liquid saturated moist packing

suspensionand particleflow pattern

shear rate γ.

τ

γ.

ττ ≠ f (σ) τ

σ

γ.

τ

normal stress σ

.τ ≈ f ( )γflow function

cubical cellpackingmodel

ϕs

εs,0= (1+ )a

d-3

particleseparation

particle volumefraction

particlefriction

a

ad < 0-0.01 <

εsπ6>

S < 1

ϕi > 30°

τ

γ =. dux dy

yx

uxdy

τ

uxdy vx

ux

τσ

a

a

τ

τ

d

d

da

a

τ

τ

d

d

τ

τ

a

a

d

d

τ

τ

a

σ

τσ

vxdy

contactcontactdeformation

Fig. 1.27



Sampling

Fig. 1.28



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Dispersity of particulate systems, About rocks, gravel ...€¦ · 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 1 1 o A 1 nm 1 µm 1 mm 1 cm 1 m wave length of visible light: