This document discusses flow behavior of particulate solids in bunkers and hoppers. It contains the following key points: 1. Mass flow and core flow are the two main types of particulate flow in hoppers. Mass flow avoids channeling while core flow can lead to channeling. 2. For mass flow hoppers, the hopper angle must be greater than the effective angle of internal friction. For core flow hoppers, channeling can be avoided by proper design based on parameters like wall friction angle, internal friction angle, and minimum slot width. 3. Bridging is another flow problem that can be avoided by ensuring minimum particle size, bulk density, hopper shape, and slot dimensions based on

1. F 4.1

2. 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
height
levels of
free bulk
surface velocity
profiles
5
7
6
4
3
2
8
1
Θ
1
7
6
8
5
4
3
2
1
Θ
ϕb
ϕb
3 3
4 4
2 2
5 5
6
7
6
7
1 1
8 8
Θ1
Θ2
plug
flow
angle of
repose
dead
zones
Θ
Channelling, Piping, Ratholing Bridging, Arching
dead
zones
ΘΘ
F 4.2

3. F 4.3

4. Dynamics of Force Balance at Cohesive Powder Bridge
B
Θ
dFT
h
Θ
W
´1
dFV
b
dFf
VF
dFV
dFG
dhB
´1
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
. a
g
dFf = Eu b l dhB
. 3 f u2 (1 - )
4 d 2
. . .
. .
...
F 4.4

5. 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
´1
´1

6. 0 10 20 30 40 50 60
45
40
35
30
25
20
15
10
5
0
hopper angle versus vertical in deg
angleofwallfrictionwindeg
Mass Flow
Core Flow
effective angle of
internal friction
e = 70°
60°
50°
40°
30°
1
2
180° - arccos
1 - sin e
2 sin e
- W - arc sin sin W
sin e
Bounds between Mass and Core Flow
axisymmetric Flow
(conical hopper)
select
F 4.6

7. 50
45
40
35
30
25
20
15
10
5
0
angleofwallfrictionwindeg
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° ande
60°
Bounds between Mass and Core Flow
Plane Flow
(wedge-shaped hopper)
F 4.7

8. max
lmin > 3 · b
min
bmin
bmin
D
lmin >3·b
min
bmin
max max
wall
b
min
- 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

9. inclined front walls
1,5 b
min
bmin
lmin > 6 · b
min
3 b
min
B
L
1max
2max
1,5 b
min
F 4.9

10. unconfinedyieldstrengthc
c,0
major principal stress during
consolidation (steady-state flow) 1
0
c = a1 · 1 + c,0
effectivewallstress'
' = 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

11. 20 30 40 50 60 70
1,5
flowfactorff
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

12. F 4.12

13. bulkdensityb
b,0
*
90°
1
1
unconfinedyieldstrengthc
1 = c,st
ff = 1
c,0
c,st
major principal stress during
consolidation (steady-state flow) 1
0
c = a1 · 1 + c,0
anglesofinternal
frictione,st,i
effectivewallstress'
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

14. Consolidation Functions of Cohesive Powders for Hopper Designbulkdensityb
b,0
*
b = b,0
* · (1 + )n
90°
effektive angle of internal friction e
1
1
unconfinedyieldstrengthc
ff = 1
c,0
c,st
major principal stress during consolidation 10
c = a1 · 1 + c,0
anglesofinternalfrictione,st,i
0 < n < 1
effectivewallstress'
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

15. F 4.15

16. F 4.16

17. (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

18. 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
Channel
A - 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

19. ct
angleofwallfriction
w
stationaryangleof
internalfriction
st
bulkdensity
ρb
mass flow hopper
core flow hopper
major principal stress 1
angleofinternalfriction
iandit
b
e st
it
i
w
uniaxialcompressivestrength
c
effectivewallstress
1
`
c
1´
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)
effectiveangleof
internalfriction
st
F 4.19

20. Calculation of Silo Pressures according to Slice-Element Method
Force Balance F = 0
Shaft (Filling F):
H
HTr
pv
pv
pn
pn pW
pW
dA
y
y
pW
pW
dydy
ph ph
H*
b · g · dy
b · g · dy
pv + dpv
pv + dpv
Hopper:
F 4.20

21. 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°
lateralpressureratioactive-plasticpassive-plasticLateral 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 +sin2
e
smooth wall w = 0
a =
1 - sin e
1 + sin e
generally 0 ≤ w ≤ e
a =
1 - sin2
w
1 +sin2
w+
(1 - sin2
w).
(sin2
e - sin2
w)-
0 = 1 - sin e soil pressure at rest
1.0
D
H
y
pw
ph
pv
pvph
pw
pw
(1 - sin2
w).
(sin2
e - sin2
w)
F 4.21

22. functionGi)
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

23. Estimation of Minimum Shaft Diameter
Process Parameters and Geometrical Apparatus Parameters
pressures p
heigthH
pW
pv
ph
shaft diameter Dmin
heightH
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 + sin2
w
+
- (1 - sin2
w).
(sin2
e - sin2
w)
(1 - sin2
w).
(sin2
e - sin2
w)
(1a) (1b)
(1c)
(2)
(3)
(4)
(5)

24. detail "Z"
maximum roof loads:
filter load: 6 kN
snow load: 1 kN /m2
gangway:
walking monoload: 1,5 kN
evenly distributed: 0,75 kN/m2
h2h3
18
d3Fl 100 x15
name and rated width of support
rated
volume
V
m3
input
outputND6
TGL0-2501
by-pass
filterlink
FTFN
reserve
workopening
levelindication
safetydevice
liftingarm
~TGL31-461
carriereye
~TGL31-343
20
40
80
100
160
320
100 200 200 600 600 150/50 200 B 160 A 300
250
300
893 x
666
B 90
B 110
B 220
B 325
A 250
-
p1 p2 p3 p4 p5 r1 r2 s1 s2 t2t1
3000
5000
3075
5080
24
36
d1) d3
numberof
bolts
workopening
20
40
80
100
160
320
rated
volume
V
m3
d1) R1 R2 1 2
[ °] [ °]
h1 h2 h4 h5 h7h3 h9
=30°
mass2)
kg
3000
3000
3000
5000
5000
3000
775 1050 35 40 325 1820 750
1750 1550 25 30 420 3200 - 900
300
200
150
350
1130
1550
2990
3480
3875
8180
300
800
2800
4290
3000
6000
11000
14000
7000
16000
5920
8920
13920
16920
11870
20870
1)
d = vessel outer diameter
2)
total mass for Al Mg 3 ( sS = 2,7 t / m3
)
=30°
Standard Silo
earthing
F 4.24

25. Comparison of Models to Calculate the
Hopper Discharge Mass Flow Rate
valid for: consider:
cohesion-
less
hopper
shape
flow
condi-
tions
airdrag
pressure
dependencyof
cohesive
F 4.25

26. F 4.26

27. 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
dischargeflowratevsinm/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

28. F 4.28

29. F 4.29

30. Equipment for Filling of Silos
- to avoid segregation
F 4.30

31. 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

32. blade
type
material installation
length in m
type
N
St
N C - 0,4 - 0,14 - N
St C - 0,4 - 0,14 - St
N C - 0,4 - 0,36 - N
St C - 0,4 - 0,36 - St
N C - 0,4 - 0,11 - N
St C - 0,4 - 0,11 - St
0,25
0,5
1,0
0,25
0,5
1,0
0,4
bended
protection
pipe
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
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 to
LS 40/A - 3,0
normal edition
LS 40/B - 0,25 to
LS 40/B - 6,0
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
ratedlength
ratedlength
F 4.32

33. Hopper Locks
horizontal gate vertical gate horizontal rotary
slide-valve
double rotary
slide-valve
ball valve rotary disk valve
discharge chute with
claw lever lock
lock with swivel chute
F 4.33

34. 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
111160
0.75
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