Prof. Dr. J. Tomas, chair of Mechanical Process …. Dr. J. Tomas, chair of Mechanical Process Engineering Fig_MPE_6 VO Mechanical Process Engineering -Particle Technology Particle

Fig. 6.1

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

Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas 02.05.2013 Figure 6.1

6. Particle interactions, powder storage and flow 6.1 Dynamics of a flowing particle packing 6.2 Fundamentals of particle adhesion and adhesion forces 6.3 Mechanics of particle adhesion 6.4 Testing methods of particle adhesion 6.5 Flow properties of cohesive powders 6.6 Testing devices and techniques of powder flow properties 6.7 Applications in silo hopper design 6.8 Evaluation of residence time distribution of processes

Fig. 6.2



Fig. 6.3



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

Physical principle

Physical assessment of product quality Particle size

d in µm Physical law Assessment char-

acteristic Value range

Evaluation

Large adhe-sion poten-tial1) FG

FH0

2

s20

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 in-tensification of adhe-sion2)

FN

FH(FN)FN

FH(FN)

( )0H

0HNH

FFFF

++⋅κ=

Contact consolida-tion coefficient κ

by flattening

0.1 – 0.3 0.3 – 0.77

> 0.77

soft very soft

extreme soft

< 10 < 1

< 0.1

Poor flow-ablity2)

σ1 σc

c

1cff

σσ

= Flow function ffc 2 - 4

1 - 2 < 1

cohesive very cohesive non-flowing

< 100 < 10 < 0.1

Large com-pres-sibility2)

σ1

∆h

n

0

st,M

0,b

b 1

σ

σ+=

ρρ

Compressibility

index n 0.05 – 0.1

0.1 - 1 compressible

very compressi-ble

< 100 < 10

Small per-meability3,4)

∆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=∆

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.

Fig. 6.4



Fig. 6.5



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: 1aa

01aa

aa

0F

FF0F

0F

0U

00U

max

max−=ε<=ε<−=

∆=ε

==

=

== (9)

Modulus of elasticity: ( )

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

Fig. 6.6



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

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.177

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

l

a

d1 d2

l

Fig. 6.7



Fig. 6.8



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):

Fig. 6.9



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 typesMoisture Bonding in a Particle Packing

Fig. 6.10



wat

er c

onte

nt X

W

XWK

Desorption

Adsorption

capillarycondensation

multimolecular layersmonolayer

relative partial pressure ϕ = pi/pSiϕK 1

2. Sorption isotherme of capillary-porous particles and packings

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 1. Capillary pressure hysteresis of a particle packing

Moisture Bonding in a Particle Packing

Fig. 6.11



Crystallisation Bridge between KCl 99 Particles d = 100 - 600 µm

Bulk caking and hardening in store house:

Fig. 6.12



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)

Fig. 6.13



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·

Particle Contact Deformation in Normal Direction without Adhesion

Fig. 6.14



Fig. 6.15



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

Fig. 6.16



Fig. 6.17



Fig. 6.18



Fig. 6.19



Fig. 6.20



(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

Fig. 6.21



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


0normal stress σ

shea

r st

ress

τyieldlocus

−σ0σ1σ2

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

Fig. 6.22



shea

r st

ress

τ

normals stress σ

Yield Locus

−σ0

0

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

−σ00

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



Fig. 6.23



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

.

Fig. 6.24



Fig. 6.25



Fig. 6.26



Fig. 6.27



displacement s

shea

r fo

rce

FS

shea

r st

ress

τ =

FS /

A

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

Fig. 6.28



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<σpreσpre

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

time t (or displacement s = vS.t)

FS

FN

σpre σ>σpre

Fig. 6.29



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

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

( )( )

( )( ) 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 sand 4 - 10 0.107 – 0.3 33 – 37 easy flowing moist fine sand 2 - 4 0.3 – 0.77 37 – 46 cohesive dry powder 1 - 2 0.77 - ∞ 46 - 90 very cohesive moist powder < 1 ∞ - non flowing,

hardened (ffct) moist powder

hydrated cement

Fig. 6.30



bulk

den

sity

ρb

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

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)

Compressibility index of powders, semi-empirical estimation for σ1 = 1 – 100 kPa

index n evaluation examples flowability 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

Fig. 6.31



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

SYL

SYL

YL

YL

free flowing

CLCL

σiso

Fig. 6.32



Fig. 6.33



Fig. 6.34



Fig. 6.35



Widely Spread Residence Time Distribution

Q3(tV)tv

Storage Time too small:

Deaeration Problems

tv

Q3(tV)

Storage Time too Large

tv

Q3(tV)

Time Consolidation Problems

Inflammation or Explosion Hazards

Deterioration Problems

Fig. 6.36



Fig. 6.37



(3)

(2)

(4)

µm ... cm ?

%o ...% ?

h ... d ?

% ?

FNFs

+ € ?

Solution for Silo Plant Design

Marketing

Cost-Benefit-Analysis

Investment Tasks

(1) Economic Goals

Layout

Flow Sheets

Storage Capacity

Selection of Main Dimensions

Moisture

Particle Size Distribution

Storage Time

Chem.-Min. Composition

Flow Behaviour Testing of Bulk Materials

Logistics

Area Requirements

Fig. 6.38



?

??

?(5) Design, Shaft and Hopper Dimension

(6) Handling Equipment Selection

Feeder (Filling)

Discharge Aids

Gate

Chute

Feeder (Discharging)

Conveyor

Mass Flow Rate

Dosage

Power (Consumption

(7) Apparatus Design and Adaption

Fig. 6.39



(8) Design of

(9) Structural Design

(10) Design and Construction, considering

Level Control Devices

Dust Collection

Safety Instrumentation

Plant and Building

Wall Thickness, SteelReinforcement

Silo Pressures

Environmental Protection

Manufacturing

Access and Cleaning

Safety

Manpower and Social Services

Environmental Protection

Repair and Maintenance

Measurement and Control Instrumentation

... etc

Fig. 6.40



Fig. 6.41



0 10 20 30 40 50 60

45

40

35

30

25

20

15

10

5

0

hopper angle versus vertical Θ in deg

angl

e of

wal

l fri

ctio

n ϕ

w in

deg

Mass Flow

Core Flow

effective angle of internal friction

ϕe = 70° 60° 50° 40° 30°

Θ ≤ 12 180° - arccos 1 - sin ϕe

2 sin ϕe

- ϕW - arc sin sin ϕW sin ϕe

Bounds between Mass and Core Flowaxisymmetric Flow

(conical hopper)

select Θ:= Θ − 3°

Fig. 6.42



50

45

40

35

30

25

20

15

10

5

0

angl

e of

wal

l fri

ctio

n ϕ

w in

deg

55

0 10 20 30 40 50 60 hopper angle versus vertical Θ in deg

Core Flow

effective angle of internal friction

ϕe = 70° 60° 50° 40° 30°

Mass Flow

Θ ≤ 60,5° +arc tan 50° -ϕe

7,73°15,07°

1-42,3° + 0,131° · exp(0,06 · ϕe)

ϕW

with ϕW < ϕ −3° and Θ ≤ e 60°

Bounds between Mass and Core FlowPlane Flow

(wedge-shaped hopper)

Fig. 6.43



Θmax

lmin > 3 · bmin

bminbmin

D

lmin>3·bmin

bmin

Θmax Θmax Θwall

bmin

- Conical Hopper (axisymmetric stress field)

Cone Pyramid

shape factor m = 1 [ 3a ]

- Wedge-shaped Hopper (plane stress field)

vertical front walls

shape factor m = 0

Θ=Θ

2tanarctan max

wall

Fig. 6.44



inclined front walls

1,5 bmin

bmin

lmin > 6 · bmin

3 bmin

B

L

Θ1max

Θ2max

1,5 bmin

( ) ( )[ ]( ) ( )

[ ]b3bBlL

lLorbBtantanarc22

maxwall

−+−

−−Θ=Θ

Fig. 6.45



Fig. 6.46



20 30 40 50 60 70

1,5

flow

fact

or

ff

effective angle of internal friction ϕe in deg

2

1

conical hopper

wedge-shaped hopper

Ascertainment of Approximated Flow Factor

(angle of wall friction ϕW = 10° - 30°)

Fig. 6.47



Fig. 6.48



Fig. 6.49



Calculation of Silo Pressures according to Slice-Element MethodForce Balance F = 0∑ ↑

Shaft (Filling F):

vFww

vFh

wF63

6363bv

bvwFv

pλtanp

pλptanλU

AHwith

HHexp1Hgρp

0gρpAUtanλ

dydp

⋅⋅ϕ=

⋅=ϕ⋅⋅

=

−−=

=⋅−⋅ϕ⋅+

H

HTr

pv

pv

pnpn pW

pWdA

y

ypW

pW

dydy

ph ph

H*

ΘΘ

ρb · g · dy

ρb · g · dy

pv + dpv

pv + dpv

Hopper:

( )

( ) ( )Θtanλtan

1m1Θtan

tan1k1mk

pktanpandpkp,tanUA2Hwith

H*HHp

H*HH

H*HH

1kHgρp

Hyforppand*HH ywith0gρypk

dydp

0gρAUtantanΘk

A1

dydAp

dydp

ww1

v1wwv1nTr

Tr

Trv0

k

Tr

Tr

Tr

TrTrbv

Trv0vTrbvv

bw1vv

⋅ϕ+=

−

ϕ++=

⋅⋅ϕ=⋅=Θ⋅

=

−+

−−

−−⋅⋅

=

==−==+⋅−

=+

ϕ+−⋅⋅+

Fig. 6.50



Fig. 6.51



Evaluation of Residence Time Distribution of Processes

1. Batch processa) Cummulative residence time distribution F(τ) b) Frequency distribution f(τ)

2. Continuous processes with ideal residence time distribution

Block flow chart:

b) ideal continuous mixer c) cascade of ideal continuous mixers

F(τ)

0 τR τ

1

f(τ)

ττR0

[ [ [ [

page 1

τ≥ττ<τ

=τΘ=τR

R

for1for0

)()(F

τ=τ∞τ≠τ

=τδ=τ=τR

R

forfor0

)()(F)(f

τ=τ

ddm

m1)(f

*

*0

*0

*

m)(m)(F

τ=τ

τ

*0m )(m* τ

0m m

1

0

τ0

tracer tracer

a) ideal plug flow channel (piston flow, ideal replacement)

3. 1st and 2nd moment of residence time distribution

( )∑∫=

−

∞

−⋅τ⋅≈ττ⋅τ===τN

1i1i,si,si,m

0,s03,1

Fillm cc

c1d)(fM

mm

a) mean residence time (complete initial moment)

b) variance of residence time distribution (2nd central moment)

( ) ( ) ( ) ( )∑∫∫=

−

∞

−⋅τ−τ⋅≈τ−τ⋅=ττ⋅τ−τ==τσN

1i1i,si,s

2mi,m

0,s

c

cs

2m

0,s0

2m3,2

2 ccc1dc

c1d)(fM)(

s

0,s

Fig. 6.52



5. Cummulative residence time distribution F(τ/τm) for cascades of ideal mixers with various stage numbers n

6. Flow channel (pipe) with axial redispersion

1

0,8

0,6

0,4

0,2

F(τ/

τ m)

0 1 2 3 4

n=1 2

∞→n3 612

τ/τm

v

x

v'

vx

m =τ

Evaluation of Residence Time Distribution of Processes page 2

1,5

1,0

0,5

0 0,5 1,0 1,5 2,0 2,5

τ m· f

(τ/τ

m)

τ/τm

3

1

∞→n

2

6n=12

∑=

ττ

−⋅

ττ

⋅−

−=τ−n

1iexp

)!1i(11)(F

n,m

1i

n,m

n = 1 ideal mixer

ideal replacement∞→n

Bodenstein number of particlesPDxvBo ⋅

=

2P 'vD ⋅Λ∝

axial dispersion (diffusion)coefficient of particles

0Bo → ideal mixer

∞→Bo ideal replacement

4. Normalized frequency distribution of residence time τm.f(τ/τm) for

cascades of ideal mixers with various stage numbers n at constant total mean residence time τm = n.τm,n = const.

with mean residence timeof one stage (unit):

nVnV

VV mn

n,mτ

=⋅

==τ

ττ

−⋅

ττ

⋅−

=τ⋅τ−

n,m

1n

n,mm exp

)!1n(n)(f

Documents

Prof. Dr. J. Tomas, chair of Mechanical Process …. Dr. J. Tomas, chair of Mechanical Process Engineering Fig_MPE_6 VO Mechanical Process Engineering -Particle Technology Particle