folien sfps 4
TRANSCRIPT
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
1Θ
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
Dynamics of Force Balance at Cohesive Powder Bridge
B
Θ
dFT
h
Θ
W
1́ dFV
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 . ag
dFf = Eu b l dhB . 3 f u2 (1 - )4 d 2
. . .
. ....
F 4.4
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
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
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
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
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
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
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
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
(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
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
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
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)
effe
ctiv
e an
gle
ofin
tern
al f
rict
ion
stF 4.19
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
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
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
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)
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
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
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
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
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
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
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