F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow...

34
F 4.1

Transcript of F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow...

Page 1: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

F 4.1

Page 2: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

Flow of Particulate Solids in Bunkers and Flow Problems

Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect (Expanded Flow)

Numbers show the sequence of discharge of bulk layers

heightlevels of free bulksurface velocity

profiles5

7

6

4

3

2

8

1

Θ

1

7

6

8

5

4

3

2

ϕb

ϕb

3 3

4 42 2

5 5

6

7

6

71 1

8 8

Θ1Θ2

plugflow

angle ofrepose

dead zones

Θ

Channelling, Piping, Ratholing Bridging, Arching

dead zones

ΘΘ

F 4.2

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F 4.3

Page 4: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

Dynamics of Force Balance at Cohesive Powder Bridge

B

Θ

dFT

h

Θ

W

1́ dFV

b

dFf

VF

dFV

dFGdhB

slot length l

Dead weight of powder bridge

Wall force

Force of inertia

Drag force of penetrating fluid

F = 0 = - dFG + dFT + dFV + dFf

dFG = b g b dhB l. . . .

dFV = 1' sin dhB cos 2l. . . .

dFT = dFG . ag

dFf = Eu b l dhB . 3 f u2 (1 - )4 d 2

. . .

. ....

F 4.4

Page 5: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

1. Mass Flow

- Avoid Channelling: Hopper angle = f(wall friction angle W, effektive angle of internal friction e) see diagrams F 4.6 and F 4.7

- Avoid Bridging:

1.1 Free Flowing Bulk Solid (avoid machanical blocking of coarse lumps or rocks):

σc,crit critical uniaxial compressive strength

ρb,crit bulk density at σ1,crit

g gravitational acceleration

article size

k = 0.6 ... 1.4 shape dependent parameter

bmin

1.2 Cohesive Powder (avoid cohesive bridges): - Effective wall stress at arch: ´ = 1/ff (2)

- Flow factor (diagram F 4.11): ff = f( e, W, ) (3)

(4)

(1a)

(1b)

Apparatus Design of Silo Hopper to Avoid Bridging

F 4.5

slot width (1c)

bmin

= + W

b · g · b

1́ 1́

Page 6: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

0 10 20 30 40 50 60

45

40

35

30

25

20

15

10

5

0

hopper angle versus vertical in deg

angl

e of

wal

l fri

ctio

n

w in

deg

Mass Flow

Core Flow

effective angle of internal friction

e = 70° 60° 50° 40° 30°

12

180° - arccos 1 - sin e

2 sin e

- W - arc sin sin W

sin e

Bounds between Mass and Core Flowaxisymmetric Flow

(conical hopper)

select

F 4.6

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50

45

40

35

30

25

20

15

10

5

0

angl

e of

wal

l fri

ctio

n

w in

deg

55

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

Core Flow

effective angle of internal friction

e = 70° 60° 50° 40° 30°

Mass Flow

60,5° +arc tan 50° - e

7,73°15,07°

1-42,3° + 0,131° · exp(0,06 · e)

W

with W 3° and e 60°

Bounds between Mass and Core FlowPlane Flow

(wedge-shaped hopper)

F 4.7

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max

lmin > 3 · bmin

b minb min

D

lmin >3·bmin

bmin

max maxwall

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

F 4.8

Page 9: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

inclined front walls

1,5 bmin

b min

lmin > 6 · bmin

3 bmin

B

L

1max

2max

1,5 bmin

F 4.9

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unco

nfin

ed y

ield

stre

ngth

c

c,0

major principal stress during consolidation (steady-state flow) 1

0

c = a1 · 1 + c,0

effe

ctiv

e w

all s

tres

s '

' = 1 / ff1

bmin 1'1'

c,crit

uniaxial compressive strength c

' c flow

' c stable arch

' c,crit

Arching/Flow Criterion of a Cohesive Powder in a Convergent Hopper

F 4.10

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20 30 40 50 60 70

1,5

flow

fac

tor

ff

effective angle of internal friction e in deg

2

1

conical hopper

wedge-shaped hopper

Ascertainment of Approximated Flow Factor

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

F 4.11

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F 4.12

Page 13: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

bulk

den

sity

b

b,0*

90°

1

1

unco

nfin

ed y

ield

str

engt

h c

1 = c,st

ff = 1

c,0

c,st

major principal stress during consolidation (steady-state flow) 1

0

c = a1 · 1 + c,0

angl

es o

f in

tern

al

fric

tion

e,

st,

i

effe

ctiv

e w

all s

tres

s

'

b,crit

b,st

' = 1 / ff1

bmin

1'

bmin,st

1'

stationary angle of internal friction st = const.

angle of internal friction i ≈ const.

effective angle of internal friction e

uniaxial compressive strength c

bulk density b

c,crit

Consolidation Functions of a Cohesive Powder for Hopper Design for Reliable Flow

F 4.13

0

Page 14: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

Consolidation Functions of Cohesive Powders for Hopper Designbu

lk d

ensi

ty

b

b,0*

b = b,0* · (1 + )n

90°effektive angle of internal friction e

1

1

unco

nfin

ed y

ield

str

engt

h c

ff = 1

c,0

c,st

major principal stress during consolidation 10

c = a1 · 1 + c,0

angl

es o

f in

tern

al f

rict

ion

e, st,

i

0 < n < 1

effe

ctiv

e w

all s

tres

s

'

e = arc sin sin st ·1 + 0

1 - sin st · 0(

bmin =(m+1) · c,crit· sin 2( w + ) b,crit · g

a1 =

c,0 =

2 · (sin st - sin i)(1 + sin st) · (1 - sin i)

2 · (1 + sin i) · sin st

(1 + sin st) · (1 - sin i)· 0

c,crit =c,0

1 - a1 · ff

b,crit

b,st

' = 1 / ff1

bmin

1'

bmin,st

1'

stationary angle of internal friction st = const.

angle of internal friction i ≈ const.

F 4.14

0

c,st

Page 15: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

F 4.15

Page 16: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

F 4.16

Page 17: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

(9a)

pv

CF

bC,min

G (angle of internal friction i or it) - function, see F 4.22

Vertical pressure at filling, F 4.20:

1 pv = f ( e, W, b, shaft cross section, silo height) (8a)

c,crit see F 4.19

a) Maximum approach at filling and consolidation:

F 4.172. Core Flow

Avoid channelling (stable funnel)

Hopper angle

2.1 Free Flowing Bulk Solid see 1.1

2.2 Cohesive Powder

Page 18: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

2. Core Flow - Supplement

Avoid channelling (stable funnel)

Hopper angle: W 2.1 Free Flowing Bulk Solid see 1.1

2.2 Cohesive Powder

bC,min

AA

ChannelA - A: Ring stress 1'' at surface of channel wall

1'' 1''

bC,min

G (Angle of internal friction i or it) - function, see F 4.22

b) Filling, consolidation and anisotropy1):

Horizontal pressure at filling, F 4.20:

1'' ph = f ( e, W, b, shaft cross section silo height)

≈ (8b)

(9b)

c) Flow and radial stress field, F 4.10, Ring stress:

(8c)1'' = 1

ffd

Flow factor of channelling:

(8d)

Two additional options:

F 4.18

Page 19: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

ct

angl

e of

wal

l fri

ctio

n

w

stat

iona

ry a

ngle

of

inte

rnal

fri

ctio

n

st

bulk

den

sity

ρb

mass flow hoppercore flow hopper

major principal stress 1

angl

e of

inte

rnal

fri

ctio

n

i and

it

b

e st

it

i

w

unia

xial

com

pres

sive

str

engt

h

c

effe

ctiv

e w

all s

tres

s 1`

c

1

1

1

1

Consolidation Functions of Cohesive Powders for Hopper Design

c,crit(core flow)

c,crit

ct,crit(mass flow)

ct,crit(core flow)

effe

ctiv

e an

gle

ofin

tern

al f

rict

ion

stF 4.19

Page 20: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

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

Shaft (Filling F):

H

HT

r

pv

pv

pnpn pW

pWdA

y

ypW

pW

dydy

ph ph

H*

b · g · dy

b · g · dy

pv + dpv

pv + dpv

Hopper:

F 4.20

Page 21: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

1.7

1.6

1.5

1.4

1.3

1.2

1.1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0

0.1

0 10 20 30 40 50 60 70 80 90

effective angle of internal friction e in deg

w=30°w=25°

w=35°

w=40°

w=45°

w=50°

w=55°

w=60°

w=65°

late

ral p

ress

ure

rati

o

ac

tive

- pl

asti

c

pas

sive

- pl

asti

cLateral Pressure Ratio = ph/pv versus Effective Angle of Internal Friction e and Angle of Wall Friction w

isostatic pressure ph = pv

w=0°

w=5°

w=10°

w=15°

w=20°

TGL 32 274/09

DIN 1055 part 6

= 0.5 0.1

e = 32° 4°

= 0.6 0.1

passive soil pressure

p =1 + sin e

1 - sin e

rough wall w = e =1 - sin2

e

1 +sin2e

smooth wall w = 0a =

1 - sin e

1 + sin e

generally 0 ≤ w ≤ e

a =1 - sin2

w

1 +sin2w+

(1 - sin2w).(sin2

e - sin2w)-

0 = 1 - sin e soil pressure at rest

1.0

D

Hy

pw

phpv

pvphpw

pw

(1 - sin2w).(sin2

e - sin2w)

F 4.21

Page 22: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

func

tion

G

i)

0 10 20 30 40 50 60 70 80

angle of internal friction i in deg

10

9

8

7

6

5

4

3

2

1

0

Function G( i) to Design a Hopper for Core Flow

F 4.22

Page 23: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

Estimation of Minimum Shaft DiameterProcess Parameters and Geometrical Apparatus Parameters

pressures p

heig

th H

pW

pv

ph

shaft diameter Dmin

heig

ht H

a) Calculation of vertical pressure

Filling /Storage

b) Consolidation function

c

1

c,0

c) Shaft design equation

D H

b

or

F 4.23

a =1 - sin2

w

1 + sin2w+

- (1 - sin2w).(sin2

e - sin2w)

(1 - sin2w).(sin2

e - sin2w)

(1a) (1b)

(1c)

(2)

(3)

(4)

(5)

Page 24: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

detail "Z"

maximum roof loads:filter load: 6 kNsnow load: 1 kN /m2

gangway:walking monoload: 1,5 kNevenly distributed: 0,75 kN/m2

h 2h 3

18d3Fl 100 x15

name and rated width of support

ratedvolume

Vm3 in

put

outp

ut N

D 6

TG

L 0

- 2

501

by-p

ass

filt

er li

nkF

TF

Nre

serv

e

wor

k op

enin

g

leve

l ind

icat

ion

safe

ty d

evic

e

lifti

ng a

rm~

TG

L 3

1 -

461

carr

ier

eye

~ T

GL

31

- 34

3

204080

100160320

100 200 200 600 600 150/50 200 B 160 A 300250

300

893 x666

B 90B 110

B 220B 325

A 250

-

p1 p2 p3 p4 p5 r1 r2 s1 s2 t2t1

30005000

30755080

2436

d1) d3

num

ber

ofbo

lts

wor

k op

enin

g

204080

100160320

rated volume

Vm3

d1) R1 R2 1 2

[ °] [ °]

h1 h2 h4 h5 h7h3 h9

=30°

mass2)

kg

300030003000

50005000

3000

775 1050 35 40 325 1820 750

1750 1550 25 30 420 3200 - 900300

200

150

350

113015502990348038758180

300800

28004290

30006000

1100014000

700016000

59208920

13920169201187020870

1) d = vessel outer diameter2) total mass for Al Mg 3 ( sS = 2,7 t / m3)

=30°

Standard Silo

earthing

F 4.24

Page 25: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

Comparison of Models to Calculate the Hopper Discharge Mass Flow Rate

valid for: consider:

cohesion-less

hoppershape

flowcondi-tions

air

drag

pres

sure

depe

nden

cy o

f

cohe

sive

F 4.25

Page 26: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

F 4.26

Page 27: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

Stationary Discharge Flow Rate versus Particle Size for Sand

conical mass flow hopper

1,5

1,4

1,3

1,2

1,1

1,0

0,9

0,8

0,7

0,6

0,5

0,4

0,3

0,2

0,1

0

disc

harg

e fl

ow r

ate

v s in

m/s

b = 0,156 m= 10°

b = 0,036 m= 10°

b = 0,036 m= 15°

b = 0,0167 m= 10°

b = 0,0103 m= 10°

calculated (Tomas)measured (Carleton)

5 10 -2 2 5 10-1 2 5 10 0 2 5 101 2 5 102

particle size d in mm

k =3, = 1, ff > 10b c

F 4.27

Page 28: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

F 4.28

Page 29: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

F 4.29

Page 30: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

Equipment for Filling of Silos

- to avoid segregation

F 4.30

Page 31: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

Methods to Control the Level of Silos

1. Pressure gauges 2. Mechanical plumb

3. Revolving blade devices

4. Membrane pressure switch

5. Conductivity measurement

6. Capacity measurement

7. Radiometric measurement 8. Ultra-sonic measurement

F 4.31

Page 32: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

bladetype

material installationlength in m

type

N

St

N C - 0,4 - 0,14 - NSt C - 0,4 - 0,14 - St

N C - 0,4 - 0,36 - NSt C - 0,4 - 0,36 - St

N C - 0,4 - 0,11 - NSt C - 0,4 - 0,11 - St

0,250,51,00,250,51,0

0,4

bendedprotection

pipe

C - 0,25 - 0,14 - NC - 0,5 - 0,14 - NC - 1,0 - 0,14 - NC - 0,25 - 0,14 - StC - 0,5 - 0,14 - StC - 1,0 - 0,14 - St

145

0,14 C

145

0,14

360

0,36

110

∅10 0,11

Revolving Blade Level Indicator LS 40

LS 40/A - 0,1 toLS 40/A - 3,0

normal edition

LS 40/B - 0,25 toLS 40/B - 6,0

with protection pipe fromcarbon (St) or stainless steel (N)

LS 40/C - 0,25 toLS 40/C - 1,0

LS 40/C - 0,4 - 0,14

installation at inclined wall

rate

d le

ngth

rate

d le

ngth

F 4.32

Page 33: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

Hopper Locks

horizontal gate vertical gate horizontal rotary slide-valve

double rotaryslide-valve

ball valve rotary disk valve

discharge chute withclaw lever lock

lock with swivel chute

F 4.33

Page 34: F 4 - Otto von Guericke University · PDF fileFlow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect ... dF f V F dF

Size in mm h1 h2 l1 l2 l3 Mass P in kg in kW

250 120 136 1245 905 180 200 315 1450 1045 217 230 400 140 1735 1235 265 260 500 119 2050 1445 317 325 630 2405 1685 380 410 800 2915 2025 465 535 1000 180 101 3530 2435 570 785

Hopper gates with drive

118

1111600.75

1.1

b1 see table above

0.55

Size in mm b1 d1 h1 h2 l1 l2 l3 Mass in kg

250 250 120 86 1097 982 180 70 315 315 1230 1115 218 92 400 410 315 140 100 1420 1305 265 123 500 515 1630 1515 318 147 630 630 1925 1810 380 221 800 800 400 160 114 2652 2362 465 393 1000 1000 180 132 3100 2810 570 570

Hopper gates

F 4.34