F 4.1
Flow of Particulate Solids in Bunkers and Flow Problems Funnel or Core Flow Mass Flow Mass Flow with Funnel Flow Effect (Expanded Flow) angle of repose ϕb ϕb
3
7
3 2 2
5
height 4 levels of free bulk 5 surface
6
6
7
7
4
6
5
velocity profiles
5 4
4 2
3 2 Θ
8
Θ1
6
plug flow
3
1 1 8
8 7
8
Θ
1
Θ2
11
Numbers show the sequence of discharge of bulk layers Channelling,
Piping, Ratholing
dead zones
Bridging, Arching
dead zones
Θ Θ
Θ
F 4.2
F 4.3
F 4.4
Dynamics of Force Balance at Cohesive Powder Bridge B
h
Θ dFT
dFG
slot length l
dh B
Θ
VF W
dFf dFV
´ dFV
1
b
´
1
F = 0 = - dFG + dFT + dFV + dFf dFG = b. g.b.dhB. l dFV =
'.sin . dhB. cos .2l
Wall force
1
dFT = dFG .
dFf =
Dead weight of powder bridge
Eu .
a g
3 . f .u2 .(1 - ) 4 .d . 2
Force of inertia
. b. l. dh
B
Drag force of penetrating fluid
Apparatus Design of Silo Hopper to Avoid Bridging F 4.5 1. Mass Flow - Avoid Channelling: Hopper angle = f(wall friction angle friction e) see diagrams F 4.6 and F 4.7
W
, effektive angle of internal
- Avoid Bridging: 1.1 Free Flowing Bulk Solid (avoid machanical blocking of coarse lumps or rocks): (1a) (1b)
slot width (1c) bmin
article size k = 0.6 ... 1.4 shape dependent parameter
1.2 Cohesive Powder (avoid cohesive bridges): - Effective wall stress at arch:
´=
- Flow factor (diagram F 4.11):
ff = f( e,
=
+
´
(2) , )
W
W
bmin
(3) (4)
b
1
/ff
1
·g·b
´
1
σc,crit
critical uniaxial compressive strength
ρb,crit g
bulk density at σ1,crit gravitational acceleration
F 4.6
Bounds between Mass and Core Flow axisymmetric Flow (conical hopper) effective angle of internal friction e = 70° 60° 50° 40° 30°
45
in deg
30
angle of wall friction
35
w
40 Core Flow
25 20 Mass Flow
15 10 5 0
0
10 20 30 hopper angle versus vertical
1 1 - sin 180° - arccos 2 2 sin
select
e e
-
40 in deg
W
50
- arc sin sin sin
W e
60
F 4.7
Bounds between Mass and Core Flow Plane Flow (wedge-shaped hopper) 55
effective angle of internal friction e = 70° 60° 50° 40° 30°
50
angle of wall friction
w
in deg
45
Core Flow
40 35 30 25 20 Mass Flow
15 10 5 0
0
10 20 30 hopper angle versus vertical
50° - e 7,73° 15,07°
40 in deg
50
60
arc tan 60,5° +
with
W
e
3° and
60°
1-
W
42,3° + 0,131° · exp(0,06 ·
e
)
F 4.8
- Conical Hopper (axisymmetric stress field) Cone
Pyramid
max
max wall
bmin
b
mi n
shape factor m = 1
-
[ 3a ]
Wedge-shaped Hopper (plane stress field) vertical front walls
D
max
b min
lm
in
>3
b min
·b
lm
in >3
mi
n
shape factor m = 0
·b
mi
n
F 4.9
inclined front walls
B L 1max
2max
1,5
b
mi
n
b min 3b
mi
n
lm
in
>6
1,5
·b
mi
n
b
mi
n
F 4.10
uniaxial compressive strength
unconfined yield strength
effective wall stress
'
c
Arching/Flow Criterion of a Cohesive Powder in a Convergent Hopper
'=
1
1
c
/ ff
c
= a1 ·
1
+
c,0
c,crit '
'
c,crit
c
flow
bmin
'
'
c
stable arch
1
c,0
0
major principal stress during consolidation (steady-state flow)
1
'
1
Ascertainment of Approximated Flow Factor (angle of wall friction
W
F 4.11
= 10° - 30°)
flow factor ff
2
1,5
conical hopper
wedge-shaped hopper 1 20
30
40 50 effective angle of internal friction
60 e in deg
70
F 4.12
F 4.13
b
Consolidation Functions of a Cohesive Powder for Hopper Design for Reliable Flow b,crit
bulk density
bulk density
b,st
* b,0
b
1 0 90°
angles of internal friction e, st, i
effective angle of internal friction
stationary angle of internal friction
angle of internal friction
st
= const.
i
≈ const.
c
1
unconfined yield strength
' effective wall stress
e
uniaxial compressive strength c
'= 1
1 / ff
= a1 ·
1
+
c
c,0
bmin
c,crit
ff = 1
'
'
c,st c,0 1
0
1
1
=
bmin,st c,st
major principal stress during consolidation (steady-state flow)
1
F 4.14
b
Consolidation Functions of Cohesive Powders for Hopper Design b,crit
bulk density
b,st
b=
*
* b,0
)n
· (1 +
b,0
0 10
discharge flow rate vs in m/s
1,5
calculated (Tomas) measured (Carleton)
1,4 1,3
b = 0,156 m = 10°
1,2 1,1 1,0
b = 0,036 m = 10°
0,9 0,8
b = 0,036 m = 15°
0,7
b = 0,0167 m = 10°
0,6 0,5 0,4 0,3 b = 0,0103 m = 10°
0,2 0,1 0 5 10
-2
2
5
10-1
2 5
10
0
2
particle size d in mm
5
101
2
5
102
F 4.28
F 4.29
Equipment for Filling of Silos
- to avoid segregation
F 4.30
F 4.31
Methods to Control the Level of Silos
1. Pressure gauges
2. Mechanical plumb
3. Revolving blade devices
5. Conductivity measurement 4. Membrane pressure switch
7. Radiometric measurement
6. Capacity measurement
8. Ultra-sonic measurement
F 4.32
Revolving Blade Level Indicator LS 40 LS 40/A - 0,1 to LS 40/A - 3,0
rated length
rated length
LS 40/B - 0,25 to LS 40/B - 6,0
normal edition with protection pipe from carbon (St) or stainless steel (N)
LS 40/C - 0,25 to LS 40/C - 1,0
LS 40/C - 0,4 - 0,14
installation at inclined wall
blade type
N
145
0,14 C
material
St 0,14
N St
0,36
N St
installation length in m
type
0,25 0,5 1,0 0,25 0,5 1,0
C - 0,25 - 0,14 - N C - 0,5 - 0,14 - N C - 1,0 - 0,14 - N C - 0,25 - 0,14 - St C - 0,5 - 0,14 - St C - 1,0 - 0,14 - St C - 0,4 - 0,14 - N C - 0,4 - 0,14 - St
145
∅10
360
110
0,11
N St
0,4 bended protection pipe
C - 0,4 - 0,36 - N C - 0,4 - 0,36 - St C - 0,4 - 0,11 - N C - 0,4 - 0,11 - St
F 4.33
Hopper Locks
horizontal gate
vertical gate
ball valve
discharge chute with claw lever lock
horizontal rotary slide-valve
double rotary slide-valve
rotary disk valve
lock with swivel chute
F 4.34
Hopper gates Size in mm b1 d1 h1 h2 l1 kg 250 250 120 86 1097 315 315 1230 400 410 315 140 100 1420 500 515 1630 630 630 1925 800 800 400 160 114 2652 1000 1000 180 132 3100
l2 982 1115 1305 1515 1810 2362 2810
l3
Mass in
180 218 265 318 380 465 570
70 92 123 147 221 393 570
Hopper gates with drive Size in mm
h1
h2
250 120 136 315 118 400 140 500 119 630 160 111 800 1000 180 101 b1 see table above
l1 1245 1450 1735 2050 2405 2915 3530
l2
l3
905 1045 1235 1445 1685 2025 2435
180 217 265 317 380 465 570
Mass P in kg in kW 200 230 0.55 260 325 0.75 410 535 1.1 785
