Product Characterization and Processing of Pharmaceutical Particulate SolidsPPPS/Fig... · 2019-03-13 · Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical

Fig_PC&P_3 Lecture Product Characterization and Processing of Pharmaceutical Particulate Solids, mechanics of cohesive particulate solids, Jürgen Tomas 13/05/2013

Figure 3.0 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering 3. Introduction into mechanics of cohesive particulate solids 3.1 Microscopic particle bonds and resulting adhesion forces 3.2 Contact mechanics of adhesive particles 3.3 Particle interaction test methods 3.4 Micro-macro transition of contact forces and bulk stresses in a particle packing 3.5 Macroscopic biaxial stress state, powder flow criteria 3.6 Product characterization test equipment and shear testing techniques 3.7 Flow and consolidation functions of cohesive particulate solids 3.8 Compression functions and shear work 3.9 Consolidation functions for hopper design 3.10 Permeation and fluidisation behaviour 3.11 Data sheet of product properties …


Figure 3.1 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering

Survey of constitutive functions, processing and handling problems of cohesive powders Property, problems

Physical prin-ciple

Physical assessment of product quality Particle size

d in µm Physical law Assessment charac-

teristic Value range

Evaluation

Large adhe-sion poten-tial1) FG

FH0

2s

20

sls,H

G

0H

dag2C

FF

⋅ρ⋅⋅π= 2

2

d)µm100(

WeightAdhesion

≈ 1 - 100

100 - 104 104 - 108

slightly adhesive adhesive

very adhesive

10 - 100 1 - 10

0.01 - 1 Large intensi-fication of adhesion2)

FN

FH(FN)FN

FH(FN)

pA

p

κ−κκ

=κ Contact consolidation

coefficient κ by flattening

0.1 – 0.3 0.3 – 0.77

> 0.77

soft very soft

extreme soft

< 10 < 1

< 0.1

Poor flowab-lity2)

σ1 σc

c

1cff

σσ

= Flow function ffc 2 - 4

1 - 2 < 1

cohesive very cohesive non-flowing

< 100 < 10 < 0.1

Large com-pressibility2)

σ1

∆h

n

0

st,M

0,b

b 1

σ

σ+=

ρρ

Compressibility in-

dex n 0.05 – 0.1

0.1 - 1 compressible

very compressible < 100 < 10

Small perme-ability3,4)

∆hW

∆hb

u b

Wf h

hku∆∆

⋅= Permeability

kf in m/s < 10-9

10-9 - 10-7 10-7 - 10-5

non-permeable very low

low

< 1 1 - 10

10 - 100

Poor fluidi-sation5,6)

( ))d(ufp P=∆ Pressure drop ∆p (Channelling)

Group C, non-fluidising

< 10

1) Rumpf, H.: Die Wissenschaft des Agglomerierens. Chem.-Ing.-Technik, 46 (1974) 1-11. 2) Tomas, J.: Product Design of Cohesive Powders - Mechanical Properties, Compression and Flow Behavior. Chem. Engng. & Techn., 27 (2004) 605-618. 3)Förster, W.: Bodenmechanik - Mechanische Eigenschaften der Lockergesteine, 4. Lehrbrief, Bergakademie Freiberg 1986. 4) Terzaghi, K., Peck, R. B., Mesri, G.: Soil mechanics in engineering practice, Wiley, New York 1996.

5) Geldart, D.: Types of Gas Fluidization, Powder Techn. 7 (1973) 285-292. 6) Molerus, O.: Fluid-Feststoff-Strömungen, Springer, Heidelberg 1982.





3.1 Microscopic particle bonds and resulting adhesion forces

Interactions between Atoms and Molecules

1. Interaction pair potential for various bonding types

ao,k a o,i ao,m a o,w ao,v

+ repulsion

atomic centre separation a

UB,V

0

UB,W

UB,m

UB,k

UB,i

VAN DER WAALS

hydrogen bridge bond

metallic

covalent

ionic

pair

pot

entia

l

U

Na+

Na+

Cl- Na+

Cl- Cl-

CC CC

C

Na+

Na+

Na+

Na+ Na+

e-

e-

e-

e-e-

HO H O

H

H..

+ - + -- + - +

+ - + -

typical crystal lattice

examplespacking

separationa0 in nm

bond energyUB in kBT*

diamondNaClNa - NaH ... OCH4

∗ θ = 25°C

aU=0 0n = 9n = 8n = 7n = 6n = 5n = 4n = 3

n = 2

n = 1

n = 0

h · ν

∆a expansion

T = 0

a0 equilibrium distance (F=0)atomic centre separation a, x

quan

tum

num

bers

UB bonding energy

h PLANCK constantν absorption frequency

Etot = En = (n + ) · h · ν (1)12

T ↑

w = exp (- ) (2) EnkBT

ψ ψ 2

x

n=2

n=1

n=0U(x) U(x)

δ2 ψ 1 δ2 ψδ x² vp

2 δ t2 - · = 0

vp2 =

h2 ν2

2m [Etot - U(x)](5) wave propagation velocity

(4) SCHRÖDINGER equation

ψ = ψmax.exp[ j(k.x - ω.t)] (3) complex wave function

2. Interaction pair potential U and thermally excited atom oscillations

3. Wave function (probability amplitude) ψ and residence frequency ψ ²

BOLTZMANNdistribution factor

energy of harmonic quantummechanical oscillator(parabolic potential curve)

0.1540.2760.3720.1760.400

290310 45 9 4

F 3.3



3. Influence of repulsive potential Ure on total interaction potential U a) ideal stiff sphere ( dA) b) compliant sphere c) compliant sphere (particle) and electric double-layer

4. Elasticity of an atomic pair at direct contact

0 0 0

0 0 0

Ure

dA dA dA

dA

UB

Ubar

a a

aacentre separation a a0

Ure ~ ( )dA a

m

m = ∞

Ure ~ ( )dA a

m

m = 9 - 12

,Ure,sphUre,el ~ exp( ) a

a37

m

∅

expansion ε = ∆a a0

MIE pair potential in a homogeneous lattice

U = - + (5)

β = Σ (7)≠i 1

α = Σ ±k1,i

pn1,i

(6)

p1,i = a1,i /a separation ratios of lattice neighbours i of contact 1 k1,i coordination numbers of packing

EM = (m - n) · n · = n · m ·α · A a0

n+3 UB

a03

(9)

atomic centre separation aaFmax

U F

tota

l pot

entia

l U

tota

l for

ce

F

F = - dUda

+ repulsion

- attraction

a0dA

U

Fmax

norm

al st

ress

σ

- pressure

- compression

εmax

+

+ tension

σmax = Fmax/A

dσdε = E

α · A β · B an am

k1,ipm

1,i≠i 1

Ure

UU

UB

UB

UB

(8) EM

ρscS,M

2= = n · m · UB,m

M

Interactions between Atoms and Molecules

centre separation a

tota

l pot

entia

l U

repu

lsio

n p

oten

tial

Ure

+ repulsion

- attraction

dσ = E · (1)

σ = = (2)

1 - εA packing density dA atomic hard sphere diameter a0 atomic packing diameter

atomic modulus of elasticity:

EM = - (3)

EM = - (4)

da a0

F 1 - εA FA εA d²A

1 - εA a0 dF εA dA

2 da a0

1 - εA a0 d²U εA dA

2 da² a0

F 3.4



Interactions of Polar Molecule Pair

Interaction pair potential due to MIE (1903) and e.g. the LENNARD-JONES potential:

UAa

Ban m= − + integer exponents n < m (1) U U

aa

aaB

U U= ⋅ ⋅

−

= =4 0

120

6

(2)

Pot. equilibrium separ.: aBAU

m n

=

−=

0

1

(3) equilibrium separation: am Bn AF

m n

=

−=

⋅⋅

0

1

(4)

Bond energy: Um n

mA

aBFn= −

−⋅

=0 (5) potential ratio:

UU

m nm

B

an aF=

=−

<0

1 (6)

Maximum attraction force: d Uda

dFda

2

2 0= − = : Fm nm

n Aa F

nmaxmax

= −−+

⋅⋅

+1 1 (7)

Separation ratios: 111

0

0

1

0

1

< =

< =

⋅ +⋅ +

=

=

−

=

−aa

mn

aa

m mn n

F

U

m n F

U

m nmax ( )

( ) (8)

Strain: ε ε εUU

FF F

F

F

aa

aa=

=

==

=

= − < = < = −00

00

01 0 1

max

max

(9)

YOUNG modulus: ( )

Ea

d Uda

m n nA

an m

UaF a F

nB

FF

= − ⋅ = − ⋅ ⋅ = ⋅ ⋅−

= =+

==

1

0

2

203

03

0

( ) (10)

Pull-off strength: σ Z

nm n

E mnm

,max =+

⋅++

+−1

111

1

(11)

-20

-15

-10

-5

0

5

10

15

20

0,00 0,10 0,20 0,30 0,40 0,50

atomic centre separation a in nm

inte

ract

ion

pair

pot

entia

l U in

10

-21 J

-20

-15

-10

-5

0

5

10

15

20

pote

ntia

l for

ce F

in 1

0-11 N

repulsion potential Uab

repulsion force Fab

attraction force Fanattraction potential Uan

aF=0aU=0

+ repulsion

- attraction aFmax

bond energy UB

total force F

total potential U



1. Shear deformation of quasi-cubical molecular packing at uniform probability of point defects flow, dV = 0

a) initial state b) uniform or simple shear y0 /a3 position shift events3

c) sharp shear crack y0 /a2 position shift events2

2. Creation of opportunities for discrete position shift events

a) no opportunity for a position shift event b) position shift events by oscillating potential curves at thermal wave transfer

centre separation a

pote

ntia

l U

centre separation a

pote

ntia

l U

∆U1

3. Opportunities for a temperature dependent distribution of various potential wells

boundary curves

a

∆WPW

= τ

· a3 (b)

(a)∆Ua

U

y

x

y0

x0

D

F'E'

AB'

y0E'

AB

C CD

E F

D

A' B'

F'

x0

x0 dx.a y0

Ey0

dx

a

γ

ux, τ

ux, τ

a

∆U1

U∆U2

Rate Dependent Flow of Monodisperse Spherical Molecules

S2

0F

0FB

aB

caa2Tk

UexpTk3

⋅

⋅π⋅λ

⋅∆

⋅⋅π⋅

=η=

=

apparent shear viscosity:

S2

0F

0FB

2B

caTa2Tk

UexpTk3

⋅⋅∆⋅α

⋅π⋅λ

⋅∆

⋅⋅π⋅=η

=

=

apparent shear viscosityfor viscoplastic flow:

F 3.6



Type Structural Model Interaction Energy U(a) in J

chemical orcovalentbond

ion bondcharge - charge

hydrogenbridge bond

charge - dipole

dipole - dipole

charge - non-polar

dipole - non-polar

non-polar - non-polar

short range, a = 0.1 - 0.2 nm1) U 60 - 350 kB · T per bond Um 150 - 800 kJ/mol

≈≈

a = 74 pm

H - H H2

directed

a = 97pm 0.1 nm≈

H

HO H2O

Q1 Q2

a

Q1 = z1 · eQ2 = z2 · enon-directed

long range a > 0.1 nm Q1 · Q2

4π ε0 · a= µi COULOMB

O O

H H

H H

H H

H H

O

Oa 0.176 nm

0.1 nm

short range a 0.3 nm≈

Ca2-

fixed dipole

freely rotating

Q · p · cos θ 4π ε0 · a²

Q² p²

6 (4π ε0)² kBT · a4

-

-

fixed dipole

rotating dipole

permanentfixed dipoles

freely rotating

p1 φ p2

aθ2θ1

p1 p2

a

Q p

aθ

Q p

a

p2 p2 [ 2cosθ1cosθ2 - sin θ1 sin θ2 cos φ] 4π ε0 · a³

p2 p2

3 (4π ε0)² kBT · a6

1 2-

1 2- KEESOM

Q² · α2 (4π ε0)² · a4

-induced

α1 α2

a

induced

p² α (1 + 3 cos² θ)2 (4π ε0)² a6

-

p² α(4π ε0)² · a6

- DEBYE

3 α1α2 hν4 (4π ε0)² · a6

- LONDON dispersion

1)1 kB·T = 4.05 · 10-21 J for T = 293 KkB = 1.38 · 10-23 JK-1 BOLTZMANN constanth = 6.628 · 10-34 Js-1 PLANCK constantε0 = 8.854 · 10-12 AsV-1m-1 permittivity of vacuume = 1.602 · 10-19 As electronic charge

p = Q · a electric dipole moment (As · m)Q electric charge (As)α electric polarizability (A²s²m²J-1)ν electronic absorption (ionization) frequencyz ion valencyµi ion potential (chemical potential)

F = - bonding forcedUda

Q αa

p α

aθ

induced

p α

a

directed

Interaction Free Energies between Atoms, Ions andMolecules in Vacuum

ISRAELACH VILI, J.: Intermolekular & Surface Forces, Academic Press London 1992, p.28

F 3.7



Molecule - plate Sphere - plate Conductor Non-conductor

Interaction Forces and Potentials between Smooth and Stiff Model Bodies

Partner Dipol moment and dispersion Electrostatic (COULOMB) (VAN-DER-WAALS)

1 a 2Sphere - sphere Point charge sphere-sphere q = 2ε0εrUel/d

d 1

d2

a

d = 2d1d2

d1 + d2

EVdW =

FVdW = -

- CH · d 24 · a

CH · d 24 a²

a1 ρn,2

EVdW = -

FVdW = -

CH · d1

12 a

CH · d1

12 a²

d 1

aAS

Q1 = π d1 ε0 εr · EQ2 = AS ε0 εr · E

UVdW = -

FVdW = -

π ρn,2 A

6 a³π ρn,2 A

2 a4

EVdW = -

FVdW = -

CH · l · d 24 2 a

√¬

√¬ 3/2

Plate - plate Conductor Non-conductorQ = AS · ε0 εr · E

EVdW =

FVdW =

- CH · AS

12 π a²

- CH · AS

6 π a3

2 Crossed cylinders

a

d1

d²

EVdW =

FVdW =

- CH· d1d2

12 a√¬

- CH · d1d2

12 a²√¬

Molecule-molecule

√¬ CH · l · d

16 2 a√¬ 5/2

Eel= ε0 εr Uel d1 lnπ2

ad1

2

l

a

d1 d2

l

a

d Mll

Eel = · q1 q2 · a l · d2ε0 εr

Eel = · q1 q2 · a AS

2 ε0 εr

a

AS

FC = · q1 q2 l · d2ε0 εr

Eel = ε0 εr · Uel1a

AS2

2

FC = ε0 εr · Uel1a2

AS2

2FC = · q1 q2

AS

2 ε0 εr

Eel = · q1 q2 · a π d1

2 ε0 εr

2

FC = · q1 q2 π d1

2 ε0 εr

2

FC = ε0 εr Uel · π2

d1a

2

2

2 Parallel chain Cylinder - cylinder Conductor Non-conductor molecules Q = π d l · ε0 εr · E

UVdW = -

FVdW = -

Α a6

6 A a7

UVdW = -

FVdW = -

3 π A l 8 dM a5

15 π A l 8 dM a6

2

2

HAMAKERconstant = f(A):

CH = π2 ρn,1 ρn,2 A

d = 2d1d2

d1 + d2

q = Q/AS =

ρn = ρ·NA/M number densitye = 1.6·10-19 A·s electronic charge ε0= 8.854·10-12 A·s/(V·m) permittivity of vacuum

nQ·eAS

(1+2· )·E surface charge densityεr,s - 1εr,s + 2

ε0≈ E electric field strengthUel electrostatic potentialF = - dU/da potential (counter) forcez ion valencyεr permittivity of interstitial medium

ISRAELACH VILI, J.: Intermolekular & Surface Forces, Academic Press London 1992, p.177SCHUBERT, H.: Handbuch der Mechanischen Verfahrenstechnik, Whiley-VCH Weinheim 2003, S.217

z1 z2 e²4π ε0 a

z1 z2 e²4π ε0 a²

π q1 q2 · d1 d22 2

2 ε0 εr (d1 + d2 + 2a)Eel =

π q1 q2 · d1 d22 2

2 ε0 εr (d1 + d2 + 2a)2FC =

UC =

FC =

FC = ε0 εr · Uell·da2

12

2

Eel = ε0 εr · Uell·da

12

2

F 3.8





Adhesion Forces between Stiff Solid Particles

a) Smooth sphere - smooth plate b) Rough sphere - smooth plate

102

10-2

1

104

10-4

10-1 1 10 102 103

particle separation a in nm

adhe

sion

forc

e F

H0

in n

N α = 20°

liquid bridgerel. humidity 50%

non-conductor

van der Waals conductor

a0 = 0.4 nm

10-7

10-9

10-5

10-10

adhe

sion

forc

e F

H0

in N

10-2 10-1 1 10 102 103 104

10-8

10-6

10-4

10-3

particle size d in µm

h r = 0

nm1 n

m5 n

m10

nm

1 µm

100 nm

FH0,VdW = . 1 + CH hr d / hr

6 a02 2.(1 + hr / a0)2

a0 = 0.4 nm molecular force equilibrium separation σlg = 72 10-3 N/m surface tensionα = 20° bridge angleθ = 0° wetting angleCH = 19 10-20 J Hamaker constant acc. to Lifschitz

.

.

qmax = 160 10-19 As/µm2 surface charge densityU = 0.5 V contact potential

CH,sls = ( CH,ss - CH,ll )2 Hamaker constant particle-water-particle

adhe

sion

forc

e F

H0

in N

particle size d in µm

10-5

10-6

10-7

10-8

10-9

non-

cond

ucto

r

liquid

bridg

e

van d

er W

aals,

dry

cond

uctor

van d

er W

aals,

wet

wei

ght o

f sph

ere

10-1 1 10 102 103

αd2

a0

hr

a0

d2

1 10 102 103

roughness height 2.hr in nm

10-5

10-6

10-7

10-8

10-9

10 µm

1 µm

adhe

sion

forc

e F

H0

in N

liquid bridge, d = 10 µm 10 µm, α = 2,5 °

d = 100 µm van der

Waals

conductor, d = 10 µ

m

non-conductor, d = 10 µm

acc. to H. Schubert (1979):

Models according to Rumpf et al. (1974):

F 3.10



Moisture Bonding in a Particle Packing

2. Liquid bridge at direct contact (a = aF=0) of two equal-sized spheres

a) Pendular state (liquid bridges)

b) Funicular state (bridges + filled pores)

c) Capillary state (filled pores)

for a real packing:

for cubic packing of monodisperse spheres:

Fs

FH

α

d/2

R1

R'2

h

R2 Fs

d/2

σlg

σlg

1. Bond types

F 3.11



wat

er c

onte

nt X

W

XWK

Desorption

Adsorption

capillarycondensation

multimolecular layersmonolayer

relative partial pressure ϕϕK 1

Sorption isotherme of a capillary-porous powder

dewatering

moisten

satu

ratio

n

capi

llary

con

dens

atio

n

adso

rptio

n

pKe

XWC water content XWXWS

capi

llary

pre

ssur

e p

K

Capillary pressure hysteresis of a particle packing

Moisture Bonding in a Particle PackingF 3.12









Liquid Bridge Bonds

Range of adsorption layers:

(9)

−⋅

⋅π⋅σ⋅

εε−

=σ⋅ 0m,W

W0A,Z

0

00 a2

aXX

d

a1

or:

( )( )

75,0

Wl

S

i75,0

ilgc X

sin1

sin18,88

d

⋅

ρρ

⋅ϕ⋅⋅ε

ϕ⋅σ⋅ε−⋅=σ

−

(10)

Range of liquid bridges:

(11)( ) ( )

( ) Wl

S

i

ilgc X

sin1d

sin2125,8⋅

ρρ

⋅ϕ−⋅⋅ε⋅ε

ϕ⋅σ⋅ε−⋅ε−⋅=σ

2

1F 3.17



( ) ( ) ( )[ ]63WEWOSs,Dt,c ttexp1XXY1 −−⋅−⋅⋅ε−⋅σ=σ

( )1tk

tkM

XM1W

W

WW

WASDst,c +⋅

⋅⋅ϑ⋅

⋅⋅ε−⋅σ=σ

t,0HNtt,H FFF +⋅κ=

s

Zst 5

t2η⋅

⋅σ⋅=κ

stit

tt,i tan/tan2

tantanϕϕ⋅κ+

ϕ=ϕ

s

sgZst,0H 5

td4F

η⋅⋅⋅σ⋅σ⋅π⋅

=

Solid Bridge Bonds

2

(13)

C hem ical reaction:

Electrostatic attraction forces

Bonds by interlocking

Magnetic attraction forces

Crystallization:

(12)

Sinter bridges: (14)

(17)

(15)

(16)

F 3.18



σct(t) Atot

σDsfAsf(t)

(1)

(2)

(3)

dt)t(dA

dt)t(dA sf

Dsfct

tot ⋅σ=σ⋅

dtV)t(dV

dt)t(d

tot

sfDsf

ct ⋅σ=σ

dtdtm

)t(dm)1()t(Lt

0 s

sf

sf

sDsfct ∫⋅

ρρ

⋅ε−⋅σ=σ

Stress Transmission at Time Consolidation (Caking)F 3.19



Micro Process Kinetics of Solid Bridge Formation by Crystal-lization

Crystallization of superaturated solution at particle contacts (YS mass re-

lated solubility):

dtdX

Ymm

mdt(t)dm

mdt(t)dm W

SW

W

s

sf

s

sf ⋅=⋅⋅

=⋅

Kinetics of water mass transfer (evaporation) from supersaturated solution

to environment:

( )WEWspWW XXAK

dtdX

−⋅⋅−=

Integration with initial condition for t = 0: XW = XW0

if ( )W63

spW Xft1AK ≠=⋅

Uniaxial compressive strength:

( ) ( )

−−⋅−⋅⋅−⋅=

63WEW0SDsct t

texp1XXYε1σσ

Flow function:

( )

−−⋅−⋅⋅

⋅=

63WEW0SDs

1ct

ttexp1XXYσ

σ2ff

Minimum hopper outlet width to avoid bridging:

( ) ( )

−−⋅

⋅−⋅⋅⋅+

=63s

WEW0SDstmin, t

texp1gρ

XXYσ1mb



Micro Process Kinetics of Solid Bridge Formation by Hydra-tion Reactions

E.g.: cement X + water

HXOHX 2W ↔⋅+ϑ

Results in hydrated (hardened) cement paste

( ) ( )dt

tdmM

Mdt

tdm WH

WW

HXHX ⋅⋅

=ϑ

( )nHWR

WH wkdt

d−= 1,

For reaction order of n = 2:


( )tk

tkM

XM

WR

WR

WW

WHXDHXct ⋅+

⋅⋅

⋅⋅

⋅−=,

,

11

ϑεσσ


( )tk

tkMg

XMmb

WR

WR

WWs

WAHXDHXt ⋅+

⋅⋅

⋅⋅⋅⋅⋅⋅+

=,

,min, 1

1ϑρ

σ



Micro Process Kinetics of Solid Bridge Formation by Viscous Contact Flow, Fusion and Sintering

Rumpf u.a. (1976): Diameter of sinter neck:

⋅

π+

σ⋅⋅

η⋅⋅

=

2Nsg

s

22

dF1

d2

5t8

dd

( ) ( ) ( ) Ntt,HOHt FttFtF ⋅κ+=

dt1

54

s

sgZs

0

0t0 ⋅η

⋅σ⋅σ⋅

εε−

⋅π⋅

=σ

s

Zst 5

t2η⋅

⋅σ⋅=κ

Superposition of adhesion forces:

( ) ( )tFFtF t,HvdWaals,Htot,H +=

Angle of internal friction:

st

it

it,i

tantan1

tantan

ϕϕ⋅κ+

ϕ=ϕ


( ))sin1()sin1(

sin)sin1(2)sin1()sin1(

tantan2

itst

t,0stit1

itst

itstct ϕ−⋅ϕ+

σ⋅ϕ⋅ϕ+⋅+σ⋅

ϕ−⋅ϕ+ϕ−ϕ⋅

=σ


( ) ( )( ) ( )[ ]1ff2sinsinsinsin1g

sin)sin1(2sin1m2b

itstitstb

t,0stitWtmin, −⋅⋅ϕ−ϕ−ϕ⋅ϕ−⋅⋅ρ

σ⋅ϕ⋅ϕ+⋅Θ+ϕ⋅+⋅=

FN

FN

d2

d



3.2 Contact mechanics of adhesive particles

Particle Contact Deformation in Normal Direction without Adhesion

rrK,el << 1

rK,plr << 1

material data: E* effective modulus of elasticity, pf micro-yield strength, ηΚ contact viscosity

hK,el

FN

FN

rrK,el

FN

hK,pl

FN

rK,pl

runloadingyield

ing

loading

WD = ∫ FR (hK) dhK

particle centre approach hK

cont

act n

orm

al fo

rce

FN

3π pf E*hK,f = ( )2r

2

kN = dFNdhK

elastic plastic andviscoplastic

force

response FR =π · r · pf · hK,pl

π · r · ηK · hK,vis·

13 E* ·√d · hK,el

3

stiffness kN = π · r · pf12 E* ·√ d · hK,el

deformation

work WD = 215 E*·√d · hK,el

5 · r · pf ·(hK,pl - hK,f)

π2

2 2

π2 · r · ηK · hK,vis · t

2·

F 3.23



Adhesive particle contacts:



3.3 Particle interaction test methods

Testing the Adhesion Force between Particle and Surface

H. Masuda and K. Gotoh, Adhesive Force of a Single Particle, pp.141, in K. Gotoh, M. Masuda, K. Higashitani, Powder Technology Handbook, Marcel Dekker, New York 1997

FHFN FH

c) Vibration method d) Impact separation method

e) Hydrodynamic method

a) Spring balance method b) Centrifugal method

u

Pressing Detachment

FC

F 3.51






Figure 3.27 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering 3.4 Micro-macro transition of contact forces and bulk stresses in a particle packing







From the Coulomb friction limit of adhesive particle contact wit h additional normal force FNV due to pre-consolidation

, (5)

by splitting in radius and centre components (index R, M)

(6)

after elimination of the angle α of contact normal

(7)

by FR → σR, FHR → σVR, FM → σM, FHM → σVM a non-linear equation of the yield locus follows:

(8)

( ) ( )[ ]α⋅⋅κ++⋅κ++⋅κ+⋅ϕ=α⋅ 2cosFFFFF1tan2sinF HRRHMM0HiR

( )( )

ϕ⋅⋅κ−ϕ⋅⋅κ−+⋅κ+⋅κ+⋅ϕ= iHRi

22HR

22MHM0HiR sinFcosFFFF1sinF

( ) ( ) ( )

ϕϕ−ϕ

−ϕ⋅ϕ

ϕ−ϕ−

σ

σ+σ−σ+σ⋅ϕϕ

⋅σ⋅ϕ=σst

ist

i2

st2

ist2

2

VR

MVM0VMi

st

VRiR cossin

tancossintan

tan

sin

( )[ ] ( ) ( )[ ]NVN0HiNVNHNiH,C,T FFF1FFFFF ⋅κ++⋅κ+⋅µ=++⋅µ=

From the Coulomb friction limit of adhesive particle contact wit h additional normal force FNV due to pre-consolidation

, (5)

by splitting in radius and centre components (index R, M)

(6)

after elimination of the angle α of contact normal

(7)

by FR → σR, FHR → σVR, FM → σM, FHM → σVM a non-linear equation of the yield locus follows:

(8)

( ) ( )[ ]α⋅⋅κ++⋅κ++⋅κ+⋅ϕ=α⋅ 2cosFFFFF1tan2sinF HRRHMM0HiR

( )( )

ϕ⋅⋅κ−ϕ⋅⋅κ−+⋅κ+⋅κ+⋅ϕ= iHRi

22HR

22MHM0HiR sinFcosFFFF1sinF

( ) ( ) ( )

ϕϕ−ϕ

−ϕ⋅ϕ

ϕ−ϕ−

σ

σ+σ−σ+σ⋅ϕϕ

⋅σ⋅ϕ=σst

ist

i2

st2

ist2

2

VR

MVM0VMi

st

VRiR cossin

tancossintan

tan

sin

( )[ ] ( ) ( )[ ]NVN0HiNVNHNiH,C,T FFF1FFFFF ⋅κ++⋅κ+⋅µ=++⋅µ=

Mikro-Makrotransition, Force and Stress Transmissionin a Sheared Particle Packing , cont.acc. to TOMAS

Usually, the stationary flow is the stressing pre-history (σVR = σR,st, σVM = σM,st):(9)

and the yield locus results in:

(10)

By a Taylor series expansion at transition to stationary flow

(11)

suitable linear equations of the yield locus follow as σR(σM) or τ(σ) functions:

(12)

(13)

)(sin 0st,Mstst,R σ+σ⋅ϕ=σ

( ) ( ) ( )

ϕ−ϕ⋅ϕ−ϕ−ϕ−

σ+σσ−σ

ϕϕ

−ϕϕ

σ+σ⋅ϕ=σ istiist2

2

0st,M

Mst,M

st

i

i

st0st,MiR sintansin

tantan1

tantansin

( ) ( )st,MMM

Rst,MMRR

st,MMdd

σ−σ⋅σσ

+σ=σσ=σσ=σ

( )

σ−

ϕσ

+σ⋅ϕ=σ+σ⋅ϕ=σ st,Mi

st,RMiZMiR sin

sinsin

( )

σ−

ϕσ

+σ⋅ϕ=σ+σ⋅ϕ=τ st,Mi

st,RiZi sin

tantan

Usually, the stationary flow is the stressing pre-history (σVR = σR,st, σVM = σM,st):(9)

and the yield locus results in:

(10)

By a Taylor series expansion at transition to stationary flow

(11)

suitable linear equations of the yield locus follow as σR(σM) or τ(σ) functions:

(12)

(13)

)(sin 0st,Mstst,R σ+σ⋅ϕ=σ

( ) ( ) ( )

ϕ−ϕ⋅ϕ−ϕ−ϕ−

σ+σσ−σ

ϕϕ

−ϕϕ

σ+σ⋅ϕ=σ istiist2

2

0st,M

Mst,M

st

i

i

st0st,MiR sintansin

tantan1

tantansin

( ) ( )st,MMM

Rst,MMRR

st,MMdd

σ−σ⋅σσ

+σ=σσ=σσ=σ

( )

σ−

ϕσ

+σ⋅ϕ=σ+σ⋅ϕ=σ st,Mi

st,RMiZMiR sin

sinsin

( )

σ−

ϕσ

+σ⋅ϕ=σ+σ⋅ϕ=τ st,Mi

st,RiZi sin

tantan

F 3.57



3.5 Macroscopic biaxial stress state, powder flow criteria



Constitutive Models for Elastic, Plastic and Viscous Material Behaviour

strain (displacement) ε = ∆l / l0

norm

al st

ress

σ

linear elasticσE

ε =non-linear elastic

anelasticelastic hysteresis

plastic hardening

shear rate (gradient) or strain rate γ = dux/dy

shea

r st

ress

τ

τ0

perfect plastic

linear viscous

shear-thickeningn > 1 (dilatant)

n = 1 linear viscoplastic

shear-thinningn < 1 (pseudoplastic)

τ = ηp. γ n + τ0

.

γ τη=

.dy τ

dux

.

perfect plastic

σ

l0∆dd0

∆llateral expansion εq

= -∆d /d0

Poisson ratio ν = -εq/ε

εpl

Flow Functions

yield stress

elastic

Uniaxial Stress-Strain-Curves

σ

ε

.ε1

.ε2>

viscoelastic

F 3.68







(1) shear and dilatancy dV > 0

cohesionτc

ϕi

0 normal stress σ

shea

r st

ress

τ

τc

yield locus

angle of internal friction

σc σc

uniaxial pressure

ϕi

0normal stress σ

shea

r st

ress

τ

yield locus

−σZ1−σZ

τc

σc

σZ1

σZ1

uniaxial tension

σZσZ

σZ σZ

isostatictension

τσ

∆h→

angle ofdilatancy ν (+)

ϕi

0normal stress σ

shea

r st

ress

τ

yield locus

σc

τc

τ c

Biaxial Stress States of Sheared Particle Packing F 3.73



σ0 σ0

σ0σ0

no deformation

isostatic tensilestrength

σiso

σisoσiso

σiso

isostatic pressure,compression dV < 0

(3) shear and compression dV < 0

(2) stationary shear dV = 0

τσ

∆h→

ν (-)angle ofdilatancy

Biaxial Stress States of Sheared Particle Packing

0normal stress σ

shea

r st

ress

τyieldlocus

−σ0 σ1σ2

ϕst

stationaryyield locus

σM,st

σR,st

stationary angle of internal friction

σ στ

ϕi

0normal stress σ

shea

r st

ess τ

yield locus

−σZσ1σ2 σM,st

σR,st

σiso

ϕi

consolidationlocusϕi ϕi

F 3.74



shea

r st

ress

τ

normals stress σ

Yield Locus

−σ00

ϕi

σM,st

Stationary Yield Locus

End point

ϕst

ϕi angle of internal friction,ϕst stationary angle of internal friction,σ0 isostatic tensile strength of unconsolidated packing; andσM,st centre stress for steady-state flow

shea

r st

ress

τ

σ1normal stress σ

τc

Yield Locus

σVRσVM

ϕi

0 σc−σZσiso


−σZ1 σ2

Consolidation Locus

σM,st

σR,st

σ1 major principal stress,σ2 minor principal stress,σc uniaxial compressive strength,σZ1 uniaxial tensile strength,σZ isostatic tensile strength,σiso isostatic pressure;

a) The three flow parameters

b) Stress states

c) Stress states at Mohr circle of steady-state flow:

shea

r st

ress

τ

normal stress σ

Yield Locus

−σ0

0

ϕi

σM,st

End point

ϕst

σ1σst

ϕst σR,st

τst

Stationary Yield Locus:

τst = cosϕst.σR,st

σst = σM,st - sinϕst.σR,st

σR,st = sinϕst.(σM,st + σ0)

Tangential point:

Yield Locus:τ = tanϕi

.(σ + σΖ)−σZ


Biaxial Stress States of Sheared Particle Packing F 3.75



σz(1) uniaxial tensile strengthσz(3) isostatic tensile strengthτc cohesionϕst stationary angle of internal frictionϕi angle of internal friction

ϕi = 0τ

τ = f ( γ ).

c) a wet-mass viscoplastic powder without Coulomb friction

σ

A preshear pointE end pointγ shear rate gradientρb bulk densityσ1 major principal stressσc uniaxial compressive strength

Yield Loci and Powder Flow Parameters for:

ϕi = ϕstϕi

Eτ

σ

a) a dry, cohesion-less or free flowing particulate solid

τ

ρb = const. EA

τc

−σZ(3) -σZ(1) σc σ1σ

ϕst

ϕi

b) a general case of moist or fine cohesive powder

.

F 3.76



3.6 Product characterization test equipment and shear testing techniques





normal stress or centre stress

direction

shea

r stre

ss or

radi

us st

ress

direct shear test triaxial test

F 3.79



load lift

twister

shear cell

yoke

powder box

control unit

FN

FS

svS = 1 - 5 mm/min

Translational Shear Cell

σ < 80 kPaτ < 70 kPa

shear cell with load yoke

F 3.80





Bulk Shear Modulus - Centre Stress Diagramof Load/Unload for Limestone Powder

0.0 5 10 15 20 25 30centre stress for steady-state flow σM,st in kPa

bulk

shea

r m

odul

us G

b in

kN

/m2

200

250

150

100

50

0.0

350

300

Gb,0

physical modelfit factor = 3.10-3

load

unload

G = 60 kN/mm²el

astic

wav

e pr

opag

atio

n ve

loci

ty c

S,b i

n m

/s

20

25

15

10

5

0.0

35

30

cS,b

( ) 0F

T

0Szb

TddF

d1

h/sddG

→→τδ

⋅⋅ε

ε−=

τ= (1)

( ) 3/1

Z23/1

22,1

V,N0Hb *E

61*Gr*E2

FF)1(3*G1G

σ⋅⋅

εε−

⋅=

⋅⋅⋅κ+⋅κ+⋅

⋅⋅ε

ε−= (2)

3/1

0

2

0

0

3/1

22,1

0H

0

00,b *E

61*Gr*E2

F3*G1G

σ⋅⋅

ε

ε−⋅=

⋅⋅⋅

⋅⋅ε

ε−= (3)

F 3.82



displacement s

shea

r fo

rce

FS

σpre

normal stress σ = FN / A

Incipient Yield and Steady-State Flow

preshearplastic yielding dV=0

instantaneousyield locus

steady-state flow

σ<σpreσpre

FN

FS

s

preshear FN

FS

s

shear

incipientyielding

0

σpre

shear dV>0

σ

shea

r st

ress

τ =

FS /

A

F 3.83



displacement s

shea

r fo

rce

FS

shea

r st

ress

τ =

FS /

A

σ2 σcσpre σ1

normal stress σ = FN / A

ϕiτc

Incipient Consolidation and Yield, Steady-State Flow

preshearplastic yielding dV=0

ϕst


ϕi

Yield Locus

Consolidation Locus

σΖ

end point

σiso σE

τE

σR,st=(σ1-σ2)/2

σM,st=(σ1+σ2)/2σ0

σ<σpre

FN

FS

s

preshear FN

FS

s

shear

shear dV>0incipientyielding

σ1 consolidation stressσc uniaxial compressive strengthϕi, ϕst angles of internal friction

steady-state flow

σpre

F 3.91



σ2 σc σ1

normal stress σ

ϕi

ϕst

σ0 σM,st=(σ1+σ2)/2

Stationary Yield Locus:

steady-state flowσR,st = sinϕst .(σΜ,st + σ0)

σΖ σiso

ϕi

Instantaneous Yield Locus:

incipient yielding

σR = sinϕi .(σΜ − σM,st) + σR,st

σR,st=(σ1-σ2)/2

τc

Consolidation Locus:σR = sinϕi .(- σΜ + σM,st) + σR,st

incipient consolidation

Constitutive Functions for Incipient Consolidation, Yield and Steady-State Flow sh

ear

stre

ss τ

0 σ

τ = tanϕi . σ - σM,st + [ ]σR,st

sinϕi

F 3.92



( )( ) ( )ist

ist1 sin1sin1

sinsin2aϕ−⋅ϕ+

ϕ−ϕ⋅=

( )( ) ( ) 0

ist

sti0,c sin1sin1

sinsin12σ⋅

ϕ−⋅ϕ+ϕ⋅ϕ+⋅

=σ

Yield Locus 1

YL 2

YL 3

YL 4

αϕst= arc sin (tanα)

σR,st= sin ϕst· (σM,st + σ0)

a) Stationary Yield Locus

radi

us st

ress

σR

,st=(

σ 1- σ

2)/2

0σ0 centre stress σM,st= (σ1 + σ2)/2

major principal stress during consolidation σ1

unia

xial

com

pres

sive

stre

ngth

σc

b) Consolidation function

ffc = 1

YL 2

YL 3

YL 4

YL 1

σ1= σc,st

σc,st

σc,0

0

F 3.93

σR,st = ast. σM,st + σR,0

σc = a1. σ1 + σc,0

ϕst= arcsin(tanα) = arcsin(ast)

σ0 = σR,0 / sin ϕst

( )( )

ϕ−⋅+

ϕ⋅+ϕ−⋅=ϕ

i1

ii1st sin1a2

sin2sin1aarcsin

(1 + sin ϕst) · (1 - sinϕi)2 · (1 + sin ϕi) · sin ϕst

σ0 = .σc,0

Experimental Determination of Cohesive Steady-State Flow Param eters of a Cohesive Powder



displacement s

shea

r fo

rce

FS

shea

r st

ress

τ

= F S

/ A

σc

σ1

σctnormal stress σ = FN / A

ϕit

ϕi

τc

Instantaneous, Stationary, Time Yield Locus and Wall Yield Locus

preshear

ϕst

t >> 0

−σ0

ϕW

steady-state flow

end point

σM,st

incipientyieldingshear

σ<σpre σ>σpreσpre σpre

stationaryyield locus

−σZ

FN

FS

s

preshearFN

timeconsolidationt >> 0 FN

FS

s

shear

FN

FS

s

wall shear

time yield locus

wall yield locusyield locus

F 3.98

time t (or displacement s = vS.t)

FS

FN



3.7 Flow and consolidation functions of cohesive particulate solids

Time Yield Loci for TiO2, Storage Time t = 24 h dS = 200 nm, Moisture XW = 0.4 %, Temperature θ = 20°C

shea

r st

ress

τ in

kPa

25

20

15

10

5

-5 0 5 10 15 20 25 30 35 40 normal stress σ in kPa

SYL

TYL 4

TYL 3

TYL 2

TYL 1

0

shea

r st

ress

τ in

kPa

15

10

5

-5 0 5 10 15 20 normal stress σ in kPa

SYL

YL 4

YL 3

YL 2

YL 10

−σ0

Instantaneous Yield Loci for TiO2

dS = 200 nm, Moisture XW = 0.4 %

F 3.99



Yield Loci and Stationary Yield Locus for TiO2

d50 = 0.61µm, XW = 0.4 %

YL 4

YL 3

CL 1

CL 2

Consolidation LocusCL 3

YL 1

-5.0 0.0 5.0 10.0 15.0 20.0 25.0−σZ -σ0

σisocentre stress σM in kPa

5.0

10.0

0.0

15.0

radi

us st

ress

σR

in k

Pa

CL 4σR,st(σM,st) σc(σ1)σR,st(σM,st) SYL

YL 2

TSC vS = 2 mm/min

-300 0.0 500 1000centre stress σM in kPa

radi

us st

ress

σR

in k

Pa

0.0

500

1000

YL 5

FO 3 CL 3

CL 2

CL 1

YL 2

YL 1

VL 4

CL 5σR,st(σM,st) SYLσR,st(σM,st) σc(σ1)

YL 4

Yield Loci and Stationary Yield Locus for TiO2

d50 = 0.61µm, XW = 27 - 30 %

σiso

PSC vS = 49 mm/min

−σZ -σ0

F 3.100



( )( )

( )( ) i

i22

i

i22

i

ic

sintan11

tan1tan11

tan11

2sin1ff

ϕ−ϕ⋅κ++

ϕ⋅κ+ϕ⋅κ++

ϕ⋅κ++

⋅ϕ−

=

Flowability assessment and contact consolidation coefficient κ(ϕi = 30°)flow

function ffc

κ-values ϕst in deg evaluation examples

100 - 10 0,01006 – 0,107 30,3 – 33 free flowing dry fine sand4 - 10 0,107 – 0,3 33 – 37 easy flowing moist fine sand2 - 4 0,3 – 0,77 37 – 46 cohesive dry powder1 - 2 0,77 - ∞ 46 - 90 very cohesive moist powder< 1 ∞ - non flowing,

hardened (ffct)moist powder

hydrated cement

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 consolidation stress σ1 in kPa

unia

xial

com

pres

sive

str

engt

h σ

c in

kPa

20.0

15.0

10.0

5.0

0.0ffc = σ1/σc 10≥

free flowing

4 < ffc < 10easy flowing

1 < ffc < 2very cohesive

σc(σ1) t = 0 σct(σ1) t = 24 h

25.0

-5.0

σ1 σc

ffc < 1hardenednon flowing

2 < ffc < 4cohesive

F 3.101Consolidation Function of Titaniaparticle size dS= 200 nm, moisture Xw= 0.4%, temperature = 20 °C



Comparison of Shear Test and Compressive Strength Testfor Time Consolidation (Caking) of Potassium Cloride K 60

0 0.1 0.2 0.3 0.4

unia

xial

com

pres

sive

stre

ngth

σ c

t in

kPa

100

80

90

70

60

50

40

30

20

10

0

moisture difference XWO - XWE in %

t = 4 . . . 6 hσ1 = 10 . . . 12 kPa

=

confidence interval

Θ 60 °C

σ

σct

τ

σ

Shear test

σ2

α

σ1 τ

Compr. strength test

σ1 = σct

σRσR = σΜ

σΜ

τσ

α

F 3.102



- 5 0.0 5 10 15 20 25 30 35 40 consolidation stress σ1 in kPa

unia

xial

com

pres

sive

str

engt

h σ

c in

kPa

20

15

10

5

0.0 ffc = σ1/σc 10≥ free flowing



25

σ1 σc

ffc < 1 non flowing

2 < ffc < 4cohesive

( )( )

( )( ) i

i22

i

i22

i

ic

sintan11

tan1tan11

tan11

2sin1ff

ϕ−ϕ⋅κ++

ϕ⋅κ+ϕ⋅κ++

ϕ⋅κ++

⋅ϕ−

=

Consolidation Function of Titania Powder

20

30

10

0.0- 4.0 0.0 5.0 10.0 15.0 20.0 25.0 normal force FN in nN

adhe

sion

forc

e F

H in

nN

40

κ = 0.77 for ffc = 2, ϕi = 30°

FH = f(FN, κ, d, E)

FH = (1 + κ) FH0 + κ FN.

Particle Adhesion Forces of Titania Particles

FN

FN FH(FN)

FH(FN)

F 3.103



Consolidation Functions of Titaniaparticle size dS = 200 nm, moisture Xw= 0.4%, temperature = 20 °C

unia

xial

com

pres

sive

and

tens

ile s

tren

gth

σc,

σ Z1 i

n kP

a

20.0

15.0

10.0

5.0

0.0ffc = σ1/σc 10≥

free flowing



25.0

- 5.0

σ1σc

ffc < 1 hardened non flowing

2 < ffc < 4cohesive

- 5.0

- 10.0

30.0σc(σ1) t = 0 σct(σ1) t = 24 hσZ1(σ1) t = 0 σZ1t(σ1) t = 24 h

σZ1

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0

consolidation stress σ1 in kPa

F 3.104



dry cohesionlessmajor principal stress at consolidation

dry cohesive

moist cohesive incompressible

moist cohesive

moist cohesiveporous compressible

Typical Consolidation Functions of Particulate Solids

dry fibrous

time consolidation, hardened

unia

xial

com

pres

sive

stre

ngth

F 3.105


Figure 3.57 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering 3.8 Compression functions and shear work

Bul

k de

nsity

ρb

ρb,0

ρb = ρb,0·(1 + )σM,stσ0

n

Centre stress of steady-state flow σM,stIsostatictensile strength -σ0

0

n = 0 incompressible

0 < n < 1 compressible

n = 1 Compressibility index of ideal gas

Compressibility index of cohesive powders for small (1 < σ < 50 kPa) and medium pressures (50 < σ < 1000 kPa)

Index n Evaluation Examples Flowabiliy 0 – 0.01 incompressible gravel free flowing

0.01 – 0.05 low compressibility fine sand 0.05 - 0.1 compressible dry powder cohesive

0.1 - 1 very compressible moist powder very cohesive

Adiabatic gas compression:

pV1

dpdV

adκ=− (1)

Isentropic powder compression:

∫∫σρ

ρ σ+σσ

⋅=ρρ st,Mb

0,b 0 0st,M

st,M

b

b dnd

(2)

Pow

der

pres

sure

σ

displace-ment ∆h

σ∆h

a) elastic

b) elastic-plastic

loading

Yunloading

elastic recovery

plastic compression

WV= σ(h) d(h/h0)

1) Uniaxial powder compression 2) Isentropic compression function

Uniaxial Compression of Cohesive Powder F 3.108



bu

lk d

ensit

y ρ

b

ρb,0

ρb = ρb,0 · (1 + )σM,stσ0

n

centre stress during consolidationor steady-state flow σM,st

isostatictensile strength -σ0

0

n = 0 incompressible

0 < n < 1 compressible

n = 1 ideal gas compressibility index

Isentropic Powder Compression

Compressibility index of powders, semi-empirical estimation index n evaluation examples flowability 0 – 0.01 incompressible gravel

0.01 – 0.05 low compressibility fine sand free flowing

0.05 - 0.1 compressible dry powder cohesive 0.1 - 1 very compressible moist powder very cohesive

Adiabatic gas compression:

pV1

dpdV

adκ=−

(1)

Isentropic powder compression:

∫∫σρ

ρ σ+σσ

⋅=ρρ st,Mb

0,b 0 0st,M

st,M

b

b dnd

(2)

F 3.109



bulk

den

sity

ρb

ρb,0

average pressure pisostatictensilestrength -σ0

0

0 < n < 1

Powder Compression

ρb = ρb,0 · (1 + )pσ0

n

com

pres

sion

rate

dρ b

/dp

ρb

p + σ0= n· dρb

dp

Wm,b = . . (1 + ) - 1pσ0

1-n n1 - n

σ0ρb,0

spec

ific

com

pres

sion

wor

k W

m,b

ρb,0

σ0n·

F 3.110



average pressure at steady-state flow σM,stisostatictensile strength -σ0

0

0 < n < 1

Compression and Preshear Work

Wm,b = . . (1 + ) - 1σM,st

σ0

1-n n1 - n

σ0ρb,0

spec

ific

com

pres

sion

and

pres

hear

wor

k W

m,b, W

m,b

,pre

displacement s

shea

r fo

rce

FS

preshear

spre

τpre, YL3

Wb, pre = ∫ FS(s) ds

τpre, YL2

τpre, YL1

FN

FS

s

Wm,b,pre= (1 + )σM,st

σ0

1-n. cosϕi

.sinϕst.spre

.σ0

hSz.ρb,0

. cosϕi.sinϕst

spre.σ0

hSz.ρb,0

F 3.111



- 5 0.0 5 10 15 20centre stress for steady-state flow σΜ,st in kPa

bulk

den

sity

ρb

in k

g/m

3

800

1000

600

400

200

0.0

ρb,0

ρb(σM,st)

−σ0

com

pres

sion

rat

e d

ρ b/d

σ M,st

in g

/J

40

50

30

20

10

0.0

dρb/dσM,st

Compression Function and Compression Rate of Titania, dS = 200 nm

- 2 0.0 5 10 15 20

centre stress for steady-state flow σΜ,st in kPa

mas

s rel

ated

pre

shea

r an

dco

mpr

essio

n w

ork

Wm

,b,p

re, W

m,b

,com

in J

/kg

4.0

5.0

3.0

2.0

1.0

0.0

Wm,b,prespre = 3 mm

Wm,b,com

Specific Preshear and Compression Work of Titania, dS = 200 nm

−σ0

v = 2 m/skinetic energy

spec

ific

pow

er c

onsu

mpt

ion

Pm

,b,p

re in

mW

/kg

40

50

30

20

10

0.0

Pm,b,pre

distortion γshea

r st

ress

τ

∫ γτ prepred

centre stress σM,st bulk

den

sity

ρb

∫ σρ

σ

)(d

st,Mb

st,M

F 3.112



Qualitative Comparisons of:

a) particle contact deformation b) particle adhesion

c) powder yield loci d) consolidation functions

Compliant and Stiff Particle Contact and Powder Behaviour

displace-ment hK

0

compliant

stiff

-FH0

forc

e F N

f) compression function

normal force FN0

compliant

stiff

adhe

sion

forc

e F H

consolidation stress σ1

0

compliantcohesive

stiff, free flowing

unia

xial

com

pres

sive

/te

nsile

stre

ngth

σc,

σ Z1

consolidation stress σ10

compliantcompressible

stiff, incompressible

bulk

den

sity

ρb

ρ b,0

−σ0

normal stress σ0

cohesive

free flowing

shea

r st

ress

τ

−σ0

SYL

SYLYLYL

e) powder constitutive models

average pressure σΜ0

cohesive

radi

us st

ress

σR

−σ0

SYLSYL

YL

YL

free flowing

CLCL

σiso

F 3.113



3.9 Consolidation functions for hopper design





3.10 Permeation and fluidisation behaviour



Fluid Flow through Particle Beds – Permeation Models

a) Fixed particle bed Valid for Model of No Author Equation

Lam

inar

flow

-aro

und

Re

< 0.

5 tr

ansi

tion

Re

= 0.

5 - 1

000

Tur

bule

nt fl

ow-a

roun

d

Re

> 10

00

Poro

sity

ε

Pore

syst

em

Ran

dom

pac

king

mon

odis

pers

e sp

here

s

Part

icle

shap

e

Remarks

1 Darcy uKukh

p

B⋅η⋅=⋅=

∆, K = Darcy constant

+ - - - + - - - Homoge-neous pore system

2 Carman, Kozeny

( )CK

2V,S3

2

B

KuA1h

p⋅⋅η⋅⋅

εε−

=∆

, KCK = Car-

man Kozeny constant, KCK = 5 for spheres of equal size with small deviation

+ - -

≈ 0.

4

+ - + - parallel bended, cylindrical pores

3 Gupte 5.5

B

K2

f Re6.5

hd

up

ε⋅=⋅

⋅ρ∆

+ - - + - + + - Dimension

analysis 4 Molerus,

Pahl, Rumpf

( ) 55.4B

K2

f Re6.514

hd

up

ε⋅⋅ε−⋅=⋅

⋅ρ∆

+ - -

0.35

– 0

.7 - + + - Basing on

results of Gupte



5 Pärnt ( )

3

2

2ST

2R

2F

hB

1d

uReh

pε

ε−⋅

⋅ψ⋅ψ⋅η

⋅⋅ξ=∆

, ε−

=1ReReh

Reh = hydraulic Reynolds number ψF = shape factor, ψR = roundness factor ξ = fluid drag coefficient

+ - - + - + - + Experiments with fine disperse particle beds

6 Burke, Plummer 3

B

K2

f

175.1hd

up

εε−

⋅=⋅⋅ρ

∆ - - + + - - - - Experiments

7 Ergun ( )3

B

K2

f

1hd

up

εε−

⋅λ=⋅⋅ρ

∆, 75.1

Re1150 +

ε−⋅=λ

+ + + + - - - - Experiments

8 Molerus

1.095.0

5.1

95.0

95.0

2

B

Re891.0

ad4.0

ad12.01

Re4

ad

21

ad692.01

Re24Eu

⋅

++

⋅+⋅+

+

⋅+⋅+⋅=

Euler number of fixed bed:

ε−ε

⋅ρ∆⋅

=1h

du3p4Eu

2

B2

fB

Re < 104

0.1

- 1

+ + + + η

ρ⋅⋅= fduRe

3

3

95.0 195.01

ad

ε−−ε−

=



b) Fluidized bed Valid for Model of No Author Equation

Lam

inar

flow

-aro

und

Re

< 0.

5 T

rans

ition

R

e =

0.5

- 100

0 T

urbu

lent

flow

-aro

und

R

e >

1000

Poro

sity

ε

Pore

syst

em

Ran

dom

pac

king

mon

odis

pers

e sp

here

s

Part

icle

shap

e

Remarks

9 Beranek, Rose, Winter-stein

( ) ( ) g1h

pfsL

B⋅ρ−ρ⋅ε−=

∆

εL = Porosity at point of bed expansion

- - + + + + + - ∆p = const. for complete range of flui-dized bed

10 Molerus

1.0

5.1

2

W

Re907.0

ad4.0

ad07.01

Re4

ad

21

ad341.01

Re24Eu

⋅++

⋅+⋅+

+

⋅+⋅+⋅=

Euler number of fluidized bed:

2f

fsW )/u(

gd34Eu

ε⋅

⋅ρ

ρ−ρ⋅= with W

W c1

Eulim=

→ε

Re < 104

0.5

- 1

+ + + + η

ρ⋅⋅= fduRe

3

3

9.0 19.01

ad

ε−−

ε−=




Figure 3.70 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering Sample No........................ Locus/Time.......................

3.11 Data Sheet of Product Properties

Author ......................................... Phone/Fax..................... ................

Apparatus.................................. Object No. .......... Pos. No........

Material..................................... Serial No......Flow Sheet No.......

Date ..............

Sheet No ........

Next Sheet No.............

0 = not, 1 = small, 2 = large 7. Processing Behaviour 13. Physicochemical Data No Characteristic Grade 1 colour sensitive No Characteristic Unit Value 1. Flow Behaviour 2 dyeing 1 solid density kg/m3 1 free flowing (> 10) 3 moisture sensitive 2 internal porosity % 2 easily flowing (10) 4 freezing 3 bulk density kg/m3 3 cohesive (4) 5 heat sensitive 4 shaking density kg/m3 4 very cohesive (2) 8. Particle Size Distribution 5 tap density kg/m3 5 hardening (< 1) Nr. size fractions ....m % 6 angle of repose deg 6 sticking, greasy 1 7 angle of wall fric. deg 7 thixotrop 2 8 BET- surface m2/g 2. Fluidization Behaviour 3 9 Blaine-surface m2/g 1 homogenous (A) 4 10 surface diam. dST µm 2 bubbling (B) 5 11 median size d50 µm 3 pistoning (D) 6 12 upper pa. size d95 mm 4 channeling (C) 7 13 low. part. size d5 µm 5 flooding 8 14 hardness (Mohs) - 6 hygroscopic 9 15 Young modulus kN/mm2 3. Compressibility 10 16 Poisson ratio - 1 compressible 11 17 tensile strength N/mm2 2 pressure-erecting 12 18 compres. strength N/mm2 3 water excreting 13 Sum Σ 19 melting point °C 4 pressure sensitive 9. Lump and Particle Shape 20 melting enthalpy kJ/kg 5 impact sensitive 1 spherical 21 spec. heat capac. kJ/kgK 6 disintegrating 2 cubic 22 therm. conductiv. W/mK 4. Abrasion Behaviour 3 needle-like 23 expansion coeff. 10-6K-1 1 hard 4 fibrous 24 net calor. value MJ/kg 2 abrasion sensitive 5 thread-like 25 inflammation tem °C 3 abrasive 6 lamellar 26 spontan. ignition °C 4 corrosive 7 sharp-edged 27 min.ignition.ener. mJ 5. Endanger Behaviour 10. Chem. Constituents dry b. 28 low. explosion li. g/kg 1 electrically chargeable 1 29 upper expl. limit g/kg 2 dusty 2 30 therm. decompos. °C 3 inflammable 3 31 soluble constitu. g/kg 4 explosive 4 32 water solubility g/l 5 out gassing 5 33 solution enthalpy kJ/kg 6 odour intensive 6 34 sp..el. resistance Ωm 7 deteriorating 7 14. Storage and Process Conditions 8 biologically active 8 1 storage capacity m3 9 cancerogen 9 Sum Σ 2 feed mass fl. rate t/h 10 toxic 11. Chem. Aggressive Constit. 3 discharge m.fl.r. t/h 11 chemically reactive 1 4 solid content g/kg 12 2 5 residence time h 13 3 6 average moisture %


Figure 3.71 Prof. Dr.-Ing. habil. J. Tomas – chair for Mechanical Process Engineering 6. Health Risks 12. Dangerous Constituents 7 min/max moisture % 1 1 8 average temp. °C 2 2 9 min/max temp. °C Sources: 10 pressure kPa p.t.o. for supplementations

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