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8/18/2019 Hopper Designing
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F 4.1
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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
profiles5
7
6
4
3
2
8
1
Θ
1
7
6
8
5
4
3
2
1Θ
ϕb
ϕb
3 3
4 42 2
5 5
6
7
6
71 1
8 8
Θ1
Θ2
plug
flow
angle of
repose
dead
zones
Θ
Channelling, Piping, Ratholing Bridging, Arching
dead
zones
ΘΘ
F 4.2
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F 4.3
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Dynamics of Force Balance at Cohesive Powder Bridge
B
Θ
dFT
h
Θ
W
1́dFV
b
dFf
VF
dFV
dFGdhB
1́
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 u
2 (1 - )
4 d 2
. . .
. ....
F 4.4
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1. Mass Flow
- Avoid Channelling:
Hopper angle = f(wall friction angle W, effektive angle of internalfriction 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
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0 10 20 30 40 50 60
45
40
35
30
25
20
15
10
5
0
hopper angle versus vertical in deg
a n g l e o f w a l l f r
i c t i o n
w
i n d e g
Mass Flow
Core Flow
effective angle of
internal friction
e = 70° 60° 50° 40° 30°
12
180° - arccos1 - sin e2 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
a
n g l e o f w a l l f r i c t i o n
w
i n d e g
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 tan50° - e7,73°
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
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max
l m i n > 3 · b
m i n
b m i n
b m i n
D
l m i n > 3 · b
m i n
bmin
max maxwall
b m i n
- 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
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inclined front walls
1 , 5 b m i n
b m i n
l m i n > 6 · b
m i n
3 b m i n
B
L
1max
2max
1 , 5 b m i n
F 4.9
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u n c o n f i n e d y i e l d s t r e n g t h
c
c,0
major principal stress during
consolidation (steady-state flow) 1
0
c = a1 · 1 + c,0
e f f e
c t i v e w a l l s t r e s s
'
' = 1 / ff 1
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
f l o w f a c t o r
f f
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
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b u l k d e n s
i t y
b
b,0
*
90°
1
1
u n c o n f i n e d y
i e l d s t r e n g t 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
a n g l e s o f i n t e r n a l
f r i c t i o n
e ,
s t ,
i
e f f e c t i v e w a l l
s t r e s s
'
b,crit
b,st
' = 1 / ff 1
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
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F 4.15
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F 4.16
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(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
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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'' = 1ff d
Flow factor of channelling:
(8d)
Two additional options:
F 4.18
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ct
a n g l e o f w a l l f r i c t i o n
w
s t a
t i o n a r y a n g l e o f
i n
t e r n a l f r i c t i o n
s t
b u l k d e n s i t y
ρ b
mass flow hopper
core flow hopper
major principal stress 1
a n g l e o f i n t e r n a l f r i c t i o n
i a n d
i t
b
e st
it
i
w
u n i a x i a l c o m p r e s s i v e s t r e n g t h
c
e f f e c t i v e w a l l s t r e s s
1`
c
1́
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)
e f f e c t i v e a n g l e o f
i n
t e r n a l f r i c t i o n
s t
F 4.19
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Calculation of Silo Pressures according to Slice-Element Method
Force Balance F = 0
Shaft (Filling F):
H
H T r
pv
pv
pnpn pW
pWdA
y
y
pWpW
d y
d y
ph ph
H*
b · g · dy
b · g · dy
pv + dpv
pv + dpv
Hopper:
F 4.20
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f u n c t i o n
G
i )
0 10 20 30 40 50 60 70 80
angle of internal frictioni
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
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Estimation of Minimum Shaft Diameter
Process Parameters and Geometrical Apparatus Parameters
pressures p
h e i g t h H
pW
pv
ph
shaft diameter Dmin
h e i g h t 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 w1 + sin2 w+
- (1 - sin2 w).(sin2 e - sin
2w)
(1 - sin2 w).(sin2 e - sin
2w)
(1a) (1b)
(1c)
(2)
(3)
(4)
(5)
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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
h 2
h 3
18d3Fl 100 x15
name and rated width of support
ratedvolume
Vm3
i n p u t
o u t p u t N D 6
T G L 0 - 2 5 0 1
b y - p a s s
f i l t e r l i n k
F T F N
r e s e r v e
w o r k o p e n i n g
l e v e l i n d i c a t i o n
s a f e t y d e v i c e
l i f t i n g a r m
~ T G L 3 1 - 4 6 1
c a r r i e r e y e
~ T G L 3 1 - 3 4 3
20
40
80
100
160320
100 200 200 600 600 150/50 200 B 160 A 300
250
300
893 x666
B 90
B 110
B 220B 325
A 250
-
p1 p2 p3 p4 p5 r1 r2 s1 s2 t2t1
3000
5000
30755080
24
36
d1) d3
n u m b e r o f
b o l t s
w o r k o p e n i n g
20
40
80
100160
320
ratedvolume
Vm3
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 - 900300
200
150
350
1130
1550
2990
34803875
8180
300
800
2800
4290
3000
6000
11000
140007000
16000
59208920
13920
169201187020870
1) d = vessel outer diameter2) total mass for Al Mg 3 ( sS = 2,7 t / m
3)
=30°
Standard Silo
earthing
F 4.24
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Comparison of Models to Calculate theHopper Discharge Mass Flow Rate
valid for: consider:
cohesion-less
hoppershape
flowcondi-tions
a i r d r a g
p r e s s u r e
d e p e n d e n c y o f
c o h e s i v e
F 4.25
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F 4.26
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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
0,60,5
0,4
0,3
0,2
0,1
0
d i s c h a r g e f l o w r a t e v s
i n 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
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F 4.29
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Equipment for Filling of Silos
- to avoid segregation
F 4.30
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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
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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
0,250,51,0
0,250,51,0
0,4
bendedprotection
pipe
C - 0,25 - 0,14 - NC - 0,5 - 0,14 - NC - 1,0 - 0,14 - N
C - 0,25 - 0,14 - StC - 0,5 - 0,14 - StC - 1,0 - 0,14 - St
1 4 5
0,14 C
145
0,14
360
0,36
110
∅ 1
0
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 from
carbon (St) or
stainless steel (N)
LS 40/C - 0,25 toLS 40/C - 1,0
LS 40/C - 0,4 - 0,14
installation atinclined wall
r a t e d l
e n g t h
r a t e d
l e n g t h
F 4.32
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Hopper Locks
horizontal gate vertical gate horizontal rotaryslide-valve
double rotaryslide-valve
ball valve rotary disk valve
discharge chute withclaw lever lock
lock with swivel chute
F 4.33
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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 260500 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