Hopper discharge rate

Transcrição

I NTERNATIONAL J OURNAL OF C HEMICAL
R EACTOR E NGINEERING
Volume 10
2012
Article A44
Micro- and Macromechanics of Hopper
Discharge of Ultrafine Cohesive Powder
Jürgen Tomas∗
Guido Kache†
∗
Otto von Guericke University, Magdeburg, [email protected]
Polysius AG, Beckum, [email protected]
ISSN 1542-6580
DOI: 10.1515/1542-6580.A44
c
Copyright 2012
De Gruyter. All rights reserved.
†
Bereitgestellt von | Otto-von-Guericke Universitaet Magdeburg
Micro- and Macromechanics of Hopper Discharge of
Ultrafine Cohesive Powder∗
Jürgen Tomas and Guido Kache
Abstract
Typical flow problems, like arching or bridging, of ultrafine cohesive powders
are caused by undesired particle adhesion, poor flowability and large compressibility and intensified by poor permeability. Thus, the physical understanding
of micromechanics of ultrafine particle flow and permeation is very essential to
design properly the product quality and to improve the process performance in
particle technology.
A force balance at a homogenously assumed, dynamic bridge is formulated at the
hopper outlet that includes the dead weight, inertia, wall and drag forces. The permeation resistance is calculated as sum of a microscopic flow-around resistance of
single particles plus the macroscopic bed resistance of the moving cohesive powder bridge. The resulting differential equation is shown for turbulent flow-through
conditions of air. The first integration gives analytical discharge velocity-time
function, steady-state discharge velocity and the characteristic discharge time of
incipient (accelerated) flow. The second integration results in analytical heighttime and velocity-height functions, discharge and residence times. This method
results in physically consistent, analytical models that are comfortable to handle
and easy to prove. The steady-state discharge velocity is compared with measurements of full-scale silos at Coperion and Zeppelin companies.
This publication describes the results of a closed collaboration between university and industry. The practical objective of the project was to solve the serious
∗
ACKNOWLEDGEMENTS The authors appreciate the financial support of the German “Arbeitsgemeinschaft industrieller Forschungsver-einigungen” (AiF, Vorhaben-Nr. 14964 BR/1). Special
thanks go to the companies sh minerals, WAM, OLI Vibrationstechnik and Schäffer Verfahrenstechnik for providing experimental material and equipment. Also we would like to thank Coperion and Zeppelin Silo- und Apparatetechnik for using testing facilities. Last but not least, a lot
of thanks got to Mr. Heinrici from Schwedes + Schulze Schüttguttechnik and all members of the
project committee for the helpful discussions and practical hints.
discharge problems of ultrafine, compressible, hardly permeable, cohesive powders by applying mechanical vibrations during their gravitational flow.
KEYWORDS: powder mechanics, accelerated vibrational flow, cohesive powder
Tomas and Kache: Hopper Discharge of Ultrafine Cohesive Powder
1
1. INTRODUCTION
Cohesive and compressible powders consist of fine (d < 100 µm), ultrafine (d <
10 µm) or nanosized particles (d < 0.1 µm). These powders show a list of flow
problems in processing and product handling equipment or in storage and transportation containers. Typical flow problems, like arching or bridging, are caused
by undesired particle adhesion or sticking. Next consequences are poor flowability, large compressibility, compactibility and intensified by poor powder permeability. Thus, the physical understanding of micromechanics of ultrafine particle
flow-around, interstitial pore flow within moving particle bed and macroscopic air
permeation of dynamic bridges during cohesive powder discharging is very essential to design properly the product quality and to improve the process performance
in particle technology.
2. MICRO-MACRO INTERACTIONS OF COHESIVE POWDER FLOW AND
DISCHARGE
During the silo hopper discharge in form of failing cohesive bridges, the cohesive
powder is expanding (dilatancy) and generates a depression within the pores and
flow channels of shear zones. Air is permeating in theses pores or channels and
the discharge of hardly permeable ultrafine powders is hindered by the air
counter-flow through the pores of the moving bed (Tomas 1991, Scheibe 1997).
Thus, the macroscopic force balance at a homogenously assumed, cohesive dynamic bridge of incremental height dh B is formulated at the outlet of convergent
hopper. That includes the dead weight of the bridge as sole driving force and as
resistances: inertia, wall and drag forces (dv/dt - acceleration, g - gravitational acceleration, ρ b - bulk density, φ W - angle of wall friction), Figure 1:
Macroscopic force balance at bridge element:
∑ F = 0 = −F
G
+ FW + FT + FF
(1)
Dead weight as sole driving force (without vertical pressure onto the bridge):
FG = ρ b ⋅ g ⋅ b ⋅ l ⋅ dhB
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Heruntergeladen am | 29.10.12 14:48
(2)
Published by De Gruyter, 2012
International Journal of Chemical Reactor Engineering
2
Vol. 10 [2012], Article A44
Wall force:
FW = sin(θ + ϕW ) ⋅ σ 1' ⋅ 2l ⋅ cos(θ + ϕW ) ⋅ dhB
(3)
Figure 1. Macroscopic force balance at cohesive homogeneous dynamic powder bridge (here a
wedge shaped hopper of unit slot length l) acc. to Tomas (1991)
Force of inertia:
FT = ρ b ⋅
dv
⋅ b ⋅ l ⋅ dhB
dt
(4)
Fluid drag force:
FF =
dp
⋅ b ⋅ l ⋅ dhB
dhB
(5)
The wall force F W stands for the frictional cohesive flow resistance and
includes the effective major principal stress at wall σ 1 ’ of radial stress field in vicinity of hopper outlet. The flow properties of the cohesive powder are expressed
by the minimum outlet width to avoid bridging b min (ff - flow factor according to
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3
Jenike’s hopper design method (Jenike 1964), m = 0 wedge-shaped and m = 1
conical hopper):
bmin =
2 ⋅ ( m + 1) ⋅ sin 2(ϕW + θ ) ⋅ (1 + sinϕi ) ⋅ sinϕst ⋅ σ 0
ρb,crit ⋅ g ⋅ [1 − sinϕst ⋅ sinϕi − (sinϕst − sinϕi ) ⋅ (2 ⋅ ff − 1)]
(6)
By a suitable micro-macro transition between microscopic particle adhesion forces and macroscopic powder stresses, these flow properties are completely
described only with three physical material parameters, see Tomas (2004, 2007):
(1) ϕ i - particle friction of failing particle contacts, i.e. Coulomb friction;
(2) ϕ st - steady-state particle friction of failing contacts, increasing adhesion
by means of flattening of contact expressed by friction angles
(sin ϕ st − sin ϕ i ) . The softer the particle contacts, the larger are the difference between these friction angles the more cohesive is the powder flowability or, in other words, the better is the compactibility;
(3) σ 0 - extrapolated isostatic tensile strength of unconsolidated powder, it
equals a characteristic cohesion force or microscopically, a characteristic
adhesion force between particle contacts without any contact deformation,
resp.
These parameters ϕ i , ϕ st and σ 0 can be back calculated from yield loci τ = f(σ)
and consolidation function σ c = f(σ 1 ) as results of powder shear tests (Tomas
2004, 2009). The critical bulk density ρ b,crit at hopper outlet is directly related to
centre stress of Mohr circle of pre-consolidation state or major principal stress σ 1
of the compressible powder.
The force balance, Eq.(1), results in a nonlinear inhomogeneous differential equation of uniformly accelerated dynamic powder bridge of incremental
height dh B (b - given hopper outlet width):
dv 2 ⋅ ( m+1) ⋅ tan θ
+
dt
b
⎡
1 ⎤ 2
dp
b
⎛ b ⎞
⋅ ⎢1 +
⋅
⋅
⋅ v = g⋅ ⎜1 − min ⎟
2 ⎥
b ⎠
⎝
⎣ 2 ⋅ ( m+1) ⋅ tan θ dhB ρ b ⋅ ur ⎦
(7)
The term (1- b min /b) describes the powder flow resistance and the inertia
effect of the cohesive dynamic bridge. Additionally, the flow-through resistance
at air permeation of cohesive bridge is considered by its pressure drop dp/dh B =
f(u r , d, d ε , ρ b , ε) and depends on the relative velocity between counter-current
fluid and bridge u r = u – v, particle size d, pore size d ε and porosity ε of the permeated and sheared particle bed as well. This dynamic cohesive powder bridge is
accelerated and slides downwards along the convergent hopper walls and generates the counter-flow of air by moving bed, internal shear zone expansion (dila-
Angemeldet | 141.44.138.14
4
tancy) and volume equivalent flow so that u r = v can be provided at the outlet
zone as the bottleneck of the silo discharge process.
2.1 Particle flow-around and powder bed permeation
The fluid drag force F F is the resistance of counter-current permeation of air
through the voids of the expanding powder bed during shear flow (dilatancy). To
calculate the permeation resistance, a physically consistent model should be used
that includes both the microscopic flow-around condition of single particle and
the macroscopic resistance of the particle bed.
To describe the drag coefficient c D of single spherical particle versus particle Reynolds number in the range of 0 < Re ≤ Re crit = 2.105, the classical threeterm model of Kaskas (1970) is preferred that includes the contributions of laminar (24/Re), transition range (4/Re0.5) and turbulent (0.4) flow-around resistances:
cD =
24
4
+
+ 0.4
Re
Re
(8)
Equivalent models of Brauer (1973), Haider and Levenspiel (1989) could be used
as well.
Next the universal model of Molerus (1993) is used to approach the pressure drop of particle bed permeation by suitable definitions of the Euler number
Eu B and the particle Reynolds number of microscopic unit cell (d ST - surface diameter, u r /ε - relative interstitial fluid flow rate of permeated pores, ρ f - fluid density, η - fluid viscosity):
ρ f 1− ε 2
dp 3
= ⋅ Eu B ( Re) ⋅
⋅
⋅ ur
dhB 4
d ST ε
Re =
u r d ST ⋅ ρ f
⋅
ε
η
(9)
(10)
A geometrical cube cell model is used to microscopically approach the
characteristic flow-through conditions within in the unit cell of the particle packing. Obviously, the relative interstitial fluid flow rate u r /ε of permeated pores is
used here to calculate physically correct the Reynolds number, The surface diameter d ST is selected to be the characteristic particle size d of a distribution. This
Angemeldet | 141.44.138.14
5
size stands for a characteristic hydraulic pore diameter in the formulation of Euler
number, Eq.(9).
d
d
a
l
l
Figure 2. Unit cell model of suspended particles in a moving bed with flow-through of fluid (Molerus 1993)
The particles may have the characteristic surface separation a to their next
neighbours. To approach the geometrically complicated flow-around and interstitial pore flow-through conditions within a moving bed (dynamic bridge) an arrangement as shown in Figure 2 is assumed (a is negative for contact deformation
by normal load). For a comfortable micro-macro transition, the characteristic ratio
d ST /a is combined with the packing density (1-ε) of the permeated particle bed
(Molerus 1993):
Vs π ⋅ d 3
π⋅d3
1− ε =
=
=
3
V
6 ⋅ l3
6 ⋅ (d + a )
(11)
3
1− ε
⎛ d ST ⎞
⎜
⎟ =
⎝ a ⎠ 0.95 0.95 − 3 1 − ε
(12)
The maximum packing density of polydisperse particle bed is selected here to be
3 (1 − ε )
max = 0.95 or (1-ε) max = 0.857.
EuB =
2
1.5
⎡d
24 ⎧⎪
1 ⎛ d ⎞ ⎤ ⎫⎪
4 ⎧⎪
⎛ d ST ⎞ ⎫⎪
⎛ d ⎞ 0.891
⋅ ⎨1 + 0.692 ⋅ ⎢ ST + ⎜ ST ⎟ ⎥ ⎬ +
⎟ ⎬ + 0.4 + ⎜ ST ⎟
⎨1 + 0.12 ⋅ ⎜
0.1
Re ⎪
2 ⎝ a ⎠ ⎦⎥ ⎪
Re ⎪⎩
⎝ a ⎠0.95 ⎪⎭
⎝ a ⎠0.95 Re
⎣⎢ a
0.95 ⎭
⎩
(13)
The total bed resistance is calculated as sum of a microscopic flow-around
resistance of single particles plus the dominant macroscopic bed resistance of the
Angemeldet | 141.44.138.14
6
porous dynamic bridge. One may consider the physical plausibility of Eqs.(12)
and (13). The limit of Euler number Eu B of expanding bed (ε → 1) during the dilatant cohesive powder flow and hopper discharge, lim⎛⎜⎝ Eu B ⎞⎟⎠ = c D , obviously reε →1
sults in the settling of single particles that includes their flow-around and drag
coefficient c D , Eq.(8), (Molerus 1993).
2.2 Discharge velocities and mass flow rate
Generally, for laminar, transitional and turbulent air flow-through conditions of
the dynamic bridge, the differential equation (7) has to be numerically solved. But
with some prerequisites analytical solutions can be obtained for laminar and turbulent air permeation as well.
Analytical solutions for laminar permeation are rather complex and will be
shown in a parallel paper. But for the sake of simplicity, the solutions of turbulent
air permeation can be briefly shown here to obtain physically obvious functions
that are easy to discuss and comfortable to handle. A step-wise constant pressure
drop dp/dh B is assumed during homogeneous air permeation through the cohesive
bridge, which does not depend so much on Re. This kind of flow-through conditions may happen when air channels are formed within the failing cohesive bridge
by dilatancy. This sliding powder bridge is expanding and their generated pore
sizes are significantly larger than the characteristic particle size. This sub-process
is not easy to model and seems to be equivalent to the group-C fluidization behaviour of a cohesive powder acc. to Geldart’s classification (Geldart 1973).
The steady-state discharge velocity v st of a cohesive bridge is obtained by
differential equation for dv/dt = 0 and results in constant mass flow rate during
discharging of convergent hopper:
⎛ b ⎞
b ⋅ g ⋅ ⎜1 − min ⎟
b ⎠
⎝
v st =
⎡
1 ⎤
b
dp
⋅
⋅
2 ⋅ ( m+1) ⋅ tan θ ⋅ ⎢1 +
2 ⎥
⎣ 2 ⋅ ( m+1) ⋅ tan θ dhB ρ b ⋅ ur ⎦
(14)
Next the differential equation (7) can be rearranged and analytically integrated
(for the sake of simplicity, a step-wise constant pressure drop is provided here)
v
t
dv
g ⎛ b ⎞
⎟ ⋅ dt ,
∫v =0 vst2 − v 2 = vst2 ⋅ ⎜⎝1 − min
b ⎠ t ∫= 0
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(15)
7
which results in the typical discharge velocity-time function of process dynamics:
⎛ t ⎞
v(t ) = v st ⋅ tanh ⎜⎜ ⎟⎟
⎝ t76 ⎠
(16)
⎛ b ⎞
b ⋅ g ⋅ ⎜1 − min ⎟
1
vst
b ⎠
⎝
t76 =
=
⎛ b ⎞
⎛ b ⎞
⎡
1 ⎤
b
dp
g ⎜1 − min ⎟ g ⎜1 − min ⎟ 2( m+1) tan θ ⋅ ⎢1 +
⋅
⋅
2⎥
b ⎠
b ⎠
⎝
⎝
⎣ 2( m+1) tan θ dhB ρ bur ⎦
(17)
The characteristic discharge time t 76 lies in a relatively small range of
tenths to seconds compared to hours as the total discharge time of a full-scale silo.
This time parameter t 76 is equivalent to 76% of the steady-state discharge velocity
v(t 76 ) = 0.76.v st . This accelerated, beginning or incipient flow is terminated nearly
at t 99 = 3.t 76 because 99.5% of the discharge velocity of steady-state flow are
achieved.
v(t99 = 3 ⋅ t76 ) = v st ⋅ tanh (3) = 0.995 ⋅ v st
(18)
With the cross-sectional area A d and powder bulk density ρ b,crit at outlet, Eq.(6),
the time dependent discharge mass flow rate m& d of incipient flow results in:
m& d (t ) = ρ b,crit ⋅ Ad ⋅ v (t )
(19)
2.3 Discharge time and residence time
The second integration leads to the analytical bridge displacement or
height-time function during the acceleration period at the hopper outlet:
⎛ t
⎜⎜
v(t)
dt
v
t
ln
cosh
=
⋅
⋅
st
76
∫t =0
⎝ t76
t
h(t) =
⎞
⎟⎟
⎠
(20)
This function includes both the beginning (accelerated) and the stationary
contribution of displacement-time function h(t) at constant discharge velocity v st ,
It is worth to note here that Eq.(20) is equivalent to physically consistent formula-
Angemeldet | 141.44.138.14
8
tions of models that describe both the accelerated single particle settling and the
zone sedimentation of nonstabilized (adhering) agglomerates in water (Olatunji
2011). In this context, the function s(t) = h 0 – h(t) is equivalent to the settling
boundary between suspended particle zone and clear water at rest. This process
dynamics can be simply checked by series experiments in cylinders with suspensions at different solid concentrations.
From the pragmatic point of view, it is very useful to extend the model by
the discharge behaviour of a completely filled hopper, i.e. dominant steady-state
discharge. The inverse discharge time-height function is obtained for a given
filled hopper height h hopper by rearranging Eq.(20):
td =
hhopper
v st
⎡
⎛ 2 ⋅ hhopper ⎞ ⎤
⎟⎟ ⎥
+ t76 ⋅ ln ⎢1 + 1 − exp⎜⎜ −
v
t
⋅
⎢⎣
st
76 ⎠ ⎥
⎝
⎦
(21)
Obviously, the first term h hopper /v st amounts to hours and is equivalent to the characteristic residence time of the cohesive powder within the steady-state plug-flow
profile of a mass flow hopper. The second term amounts to tenths or seconds and
shows the contribution of the beginning or accelerated hopper flow. This last term
is important to obtain the required uniform discharge velocity without oscillations
by fluctuating powder flow and permeation properties. Replacing the discharge
time t d in the tanh-function of Eq. (16) by this Eq.(21), the characteristic discharge velocity-height function is obtained as position dependent cohesive powder bridge velocity:
⎛ 2⋅h ⎞
⎟⎟
v ( h ) = v st ⋅ 1 − exp⎜⎜ −
⎝ v st ⋅ t76 ⎠
(22)
In this context, the discharge time-height function, Eq.(21), is extended by the
filled hopper volume V hopper :
td =
⎡
⎛ 2 ⋅ Vhopper ⎞ ⎤
⎟⎟ ⎥
+ t76 ⋅ ln ⎢1 + 1 − exp⎜⎜ −
Ad ⋅ v st
A
v
t
⋅
⋅
⎢⎣
d
st
76 ⎠ ⎥
⎝
⎦
Vhopper
(23)
Eq.(23) consists of the first steady-state discharge term and the second
term for beginning, incipient or accelerated flow. For comparatively large filling
volume V hopper , fast kinetics or small time parameter t 76 and for small steady-state
Angemeldet | 141.44.138.14
9
discharge velocity v st the last term in Eq.(23) can be simplified. For
Vhopper /( Ad ⋅ v st ⋅ t76 ) > 2 and consequently 1 − exp( −4) = 0.98 ≈ 1 follows:
td ≈
Vhopper
Ad ⋅ v st
+ t76 ⋅ ln 2
(24)
The first term is equivalent to the averaged residence time, with tV during
steady-state discharge of a mass flow hopper. This mass flow pattern is equivalent
to plug-flow and its uniform residence time distribution
Vhopper
Ad ⋅ v st
=
Vhopper
= tV
V&
(25)
st
and the total residence time t V = t d includes the short acceleration period of dynamic bridge as well:
tV ≈ tV + t76 ⋅ ln 2
(26)
Finally, this is the proof of physical plausibility of laborious derivation of
these analytical models which are comfortable to handle - q.e.d. All these equations were derived for a physically consistent model-based data evaluation of
hopper discharge tests and to calculate the discharge velocities.
3. FULL-SCALE SILO DISCHARGE TESTS
To prove practically the results from both the lab shear tests and models, experiments on the full-scale silos are carried out. Therefore the testing facilities of the
companies Coperion and Zeppelin Silo- und Apparatetechnik are used, see Figure
3 and Table 1.
The discharge of the selected ultrafine cohesive limestone powder from
two different silos is investigated. During the experiments the frequency and amplitude of oscillations, and the annular gap width b gap are varied. Additionally,
different operation times of the vibrating hopper are investigated. The vibrating
hoppers are provided by the companies WAM and Schäffer Verfahrenstechnik.
Figure 3 shows a scheme of the test set-up during the full-scale silo tests. The vibrating hopper is mounted to the silo taking a hopper interface. By the vibrating
hopper sinusoidal oscillations are applied on the powder within the silo. The vi-
Angemeldet | 141.44.138.14
10
brating hopper is driven by an unbalanced motor. By adjusting the unbalanced
mass the amplitude and frequency of oscillations can be changed.
silo
load cells
hopper
interface
f
ae
acceleration
sensors
&
m
f
a
vibrating
e
hopper
Figure 3. Scheme including geometrical data of test silos at Coperion and Zeppelin companies
To determine the mass flow rate the silo is bedded on load cells and the total weight of the silo is continuously measured. The oscillation acceleration is
continuously measured, too. Four accelerations sensors are mounted on the silo
test rigs. In case of test rig “A”, there are two sensors on the vibrating hopper, one
sensor on the hopper interface and one sensor inside the silo within the powder.
But unfortunately, the support breaks during a discharge test and the internal sensor was destroyed and discharged with flowing powder. The powder is discharged
from the silo hopper into a flexible intermediate bulk container (FIBC) – so-called
big bag. Additionally, at test rig “B” two sensors are mounted on the vibrating
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11
hopper and two sensors on the hopper interface. However here, the powder is
filled into a second silo by pneumatic conveying and not into big bags.
Table 1. Geometrical data of test silos
Property
test rig „A“
test rig „B“
Volume of silo V in m³
3.3
30.0
Silo diameter D in mm
1050
2400
Hopper angle θ in deg
30
30
Diameter D of vibrating hopper in mm
600
1300
Outlet diameter b outlet of vibrating hopper in mm
273
273
323
Annular gap widths b gap in mm
40
90
160
170
90
160
253
325
Used silo volume V in m³
2.5
25
3.1 Hopper dimensions
The powder flow characteristics like major principal stress σ 1 , uniaxial compressive strength σ c , and effective angle of internal friction φ e are determined by a vibrating shear tester (Kollmann 2002).
According to the powder flow characteristics, the critical hopper dimensions are calculated using Jenike’s design method. The outlet dimensions of the
selected vibrating hoppers were b outlet ≤ 323 mm and b gap ≤ 325 mm. But according to above shear test results discharge width b min = 903 mm (conical outlet) and
b min = 452 mm (annular gap) are necessary to avoid bridging at gravitational discharge without applying vibrations, Figure 4. Thus, bridging happens in the silos,
which has been confirmed by the full-scale tests.
With increasing vibration velocity the minimum outlet width to avoid
bridging decreases and the maximum hopper angle to obtain mass flow increases,
Figure 4 (Kache 2010a and 2010b).
Angemeldet | 141.44.138.14
12
conical outlet
angle conical hopper
wedge shape outlet / annular gap
angle wedge shaped hopper
1000
900
30
outlet width bmin in mm
800
25
700
600
20
500
15
400
300
10
200
5
100
maximum hopper angle Θ max in degree
35
0
0
0
0,05
0,1
0,15
0,2
0,25
0,3
peak oscillation velocity vpeak in m/s
Figure 4. Minimum outlet width b min to avoid bridging and maximum hopper angle Θ max to obtain
mass flow versus peak oscillation velocity v peak of cohesive limestone powder (Kache 2010a and
2010b)
3.2 Discharge velocity and mass flow rates
There are two bottlenecks for powder flow within vibrating hoppers, the
annular gap b gap inside the hopper and the outlet width b outlet , Figure 5. The vibrating hopper can be overwhelmed with powder if the mass flow rate of the annular
gap m& I is larger than the mass flow rate through the outlet m& II . If the complete
volume of the vibrating hopper below the annular gap V hopper is filled, an undesired consolidation of the powder occurs by the applied vibrations. To prevent this
undesired consolidation and compression, vibrating hoppers have to be operated
with idle periods.
1st step annular gap:
m& I = ρ b, gap ⋅ Agap ⋅ v gap
Angemeldet | 141.44.138.14
(27)
13
2nd step outlet:
m& II = ρ b,outlet ⋅ Aoutlet ⋅ voutlet
(28)
limiting condition of mass flow rates to avoid undesired consolidation within
hopper V hopper :
m& I ≤ m& II
(29)
vgap
&I
m
vgap
V hopper
bgap
voutlet
Θ
m& II
boutlet
Figure 5: Scheme of critical dimensions and discharge mass flow rates of the vibrating hopper
Eqs.(10), (12), (13) and (14) are used to calculate the steady-state discharge velocity. Within the model the minimum outlet width b min considers the
powder flowability tested by the shear cell (Scheibe 1997, Kache 2010a and
2010b). The given hopper dimensions are included by the hopper angle θ and the
outlet width b (m = 0 wedge-shaped, m = 1 conical hopper). During the discharge,
the powder is expanding (dilatancy) and generates a depression within the pores
and flow channels of shear zones. Air is permeating in theses pores or channels
and the discharge of fine powders is hindered by the air counter-flow through the
pores of the moving bed (Scheibe 1997, Kache 2010a and 2010b) . This resistance
of powder discharge is determined by the fluid drag (pressure drop) of the moving
bed expressed by the Euler number Eu B , Eq.(13). This Euler number characterises
inversely the permeability of the powder and increases strongly with decreasing
particle or pore size d ε , respectively and porosity ε. The experimental results and
Angemeldet | 141.44.138.14
14
model calculations, Eq.(14), are compared in Figure 6 for limestone powder. The
cross-sectional areas of the annular gap and hopper outlet, resp., are calculated as:
(
2
Agap = π ⋅ D ⋅ bgap − bgap
Aoutlet =
)
(30)
π 2
⋅ boutlet
4
(31)
The steady-state velocity of the powder within the annular gap is calculated taking the measured mass flow rates and the dimensions of the vibrating
hopper, Eqs.(27) and (28). A constant bulk density ρ b is assumed at the annular
gap.
steady-state discharge velocity vst,gap in m/s
0.03
test rig A
test rig B
model (test rig A)
model (test rig B)
II
I
0.025
0.02
0.015
0.01
0.005
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
cross-sectional area of annular gap Agap in m²
0.8
0.9
1
Figure 6. Measured and calculated steady-state discharge velocities v st,gap versus the crosssectional area of annular gap A gap of cohesive limestone powder (Kache 2010a)
According to Eqs.(30) and (31), there are two steady-state discharge velocities for the annular gap and the outlet; these are the maximum velocities and
also maximum mass flow rates. The velocity within the gap depends on the gap
width; the velocity at the outlet is independent from the gap width. If this maximum velocity at the outlet is reached, a bigger annular gap does not lead to larger
Angemeldet | 141.44.138.14
15
mass flow rate. However, the velocity within the annular gap is limited by the
geometrical condition at the outlet. This is shown in Figure 6. At first, for the
cross-sectional area of annular gap A gap < 0.08 m² (test rig A) and A gap < 0.15 m²
(test rig B) bridging occurs, respectively. The steady-state discharge velocity is
zero v gap = 0. Thus values for bridging are depending on the applied level of the
peak oscillation velocity, see Figure 5.
Next the discharge velocity within the annular gap increases to a maximum and then the velocity decreases, because the maximum velocity at the outlet
is reached, the volume below the annular gap is filled and the powder flow in the
gap is slowed down. For this reason the annular gap governs the powder flow in
region I; afterwards the maximum velocity on the conical outlet dominates the
powder discharge (region II). Undesired consolidation of the powder can occur if
the mass flow rate through the annular gap is higher than the mass flow rate
through the outlet.
To avoid this undesired effect the operation and idle times of a vibrating
hopper can be adjusted, if the limit for the minimum outlet width is reached. During the idle time bridging occurs on the annular gap and the mass flow rate is
completely stopped or at least slowed down.
To calculate the time to fill the hopper volume below annular gap, a mass
balance for this stored volume V Hopper is considered.
d (ρ b ⋅ Ψ )
dm
= Vhopper
= mI − mII
dt
dt
(32)
Up to the maximum operation time t op , undesired consolidation does not
occur, provided that the bulk density ρ b is assumed to be constant:
t op = Ψmax ⋅
ρ b ⋅Vhopper
m& I − m& II
(33)
Additionally, the necessary time to empty this volume can be calculated idle time or period without excitation of vibrations t idle . Therefore it is assumed,
that powder flow through the annular gap is immediately stopped after switch-off
the vibrations:
tidle = Ψmax ⋅
ρ b ⋅Vhopper
m& II
Angemeldet | 141.44.138.14
(34)
16
The maximum degree of filling is given by Ψ max . For safe operation without undesired consolidation Ψ max = 80% should be aspired.
At the full-scale tests, the operation time intervals have been found to be
between 20 s to 30 s and the idle periods between 40 s to 80 s. These model calculations results in 30 s for operation and 60 s for idle of vibrating hopper. That results in a sufficient agreement of practical experiences. But never the less, the fit
between calculated and real mass flow rates should be improved to avoid too
large differences between model calculation and the problematic discharge and
dosing behaviour of cohesive ultrafine powders in hoppers.
4. CONCLUSIONS
Concerning the described modelling of an incremental bridge-element, for the
sake of simplicity and to get physically consistent and comfortable analytical solutions, only homogeneous material functions (e.g. step-wise constant permeability) were used to model the accelerated cohesive bridge discharge. To avoid these
simplifications, next, combined DEM and CFD simulations with size-dependent
particle properties, soft contact models with history dependent adhesion forces
and thus, spatial and time-variant material functions will be accomplished.
Due to the mechanical vibrations, ultrafine cohesive powder discharge is
enabled and influenced by the maximum vibration velocity. The discharge velocity can be controlled by pulsed operation of the vibrating hopper. In contrast, continuous operation of the vibration hopper leads to a decrease of discharge velocity
and mass flow rate. It is possible to predict necessary operating and idle periods
(pulsed operation) by calculation of the discharge velocities at the annular gap of
a vibrating hopper and at the outlet.
NOTATION
a
A
b
b min
cD
d
D
d ST
Eu
F
g
particle surface separation, µm
area, m²
given hopper outlet width, m
minimum hopper outlet width to avoid bridging, m
drag coefficient
particle size, µm
silo diameter, m
particle surface (Sauter) diameter, µm
Euler number
force, kN
gravitational acceleration, m/s²
Angemeldet | 141.44.138.14
h
h0
l
m
m&
p
Re
t
u
v
V
17
height, m
initial suspension height, m
length, m
hopper shape parameter
powder mass flow rate, kg/s
fluid pressure
particle Reynolds number
time, s
fluid velocity, m/s
discharge velocity (flow rate) of powder bridge, m/s
silo or hopper volume, m³
Greek letters
ε
η
θ
ρ
σc
σ1
σ1’
σ0
τ
φi
φ st
φW
ψ
porosity
fluid viscosity, Pa.s
hopper angle, deg
density, kg/m³
uniaxial compressive strength, kPa
major principal stress, kPa
effective major principal stress at hopper wall, kPa
isostatic tensile strength of unconsolidated powder, kPa
shear stress, kPa
angle of internal friction, deg
stationary angle of internal friction, deg
wall friction angle, deg
hopper volume filling degree
Subscripts
b
B
crit
d
e
ε
F
gap
G
i
idle
bulk
bridge
critical
discharge
effective
pore
fluid drag
ring gap
gravitation
internal
idling
Angemeldet | 141.44.138.14
18
max
min
op
r
st
T
V
W
x
76
maximum
minimum
operation
relative
stationary
inertia
residence
wall
average
0.76 of terminal value
REFERENCES
Brauer, H., “Stoff-, Impuls- und Wärmetransport durch die Grenzfläche
kugelförmiger Partikel”, Chemie Ingenieur Technik, 1973, 45, 1099-1103.
Geldart, D., “Types of Gas Fluidization”, Powder Technology, 1973, 7, 285-292.
Haider, A. and O. Levenspiel, “Drag coefficient and terminal settling velocity of
spherical and nonspherical particles”, Powder Technology, 1989, 58, 6370.
Jenike, A. W., “Storage and Flow of Solids”, 1964, Bulletin of the University of
Utah.
Kache, G., “Verbesserung des Schwerkraftflusses kohäsiver Pulver durch
Schwingungseintrag”, 2010a, (Ph.D. thesis) docupoint, Magdeburg.
Kache, G. and J. Tomas, “Ausfließen eines kohäsiven, hochdispersen Pulvers”,
Schüttgut, 2010b, 16, 246-252.
Kaskas, A.A., “Schwarmgeschwindigkeit in Mehrkornsuspensionen am Beispiel
der Sedimentation“, 1970, Diss. (Ph.D. thesis) TU Berlin.
Kollmann, T. and J. Tomas, “Effect of Applied Vibrations on Silo Hopper Design”, Particulate Science & Technology, 2002, 20, 15-31.
Molerus, O., “Principles of Flow in Disperse Systems”, 1993, Chapman & Hall,
London.
Olatunji, O. N. and J. Tomas, “Flow-around and flow-through dynamics of particle sedimentation - micro and macro behaviour, 8th European Congress of
Chemical Engineering (ECCE8), 2011, Berlin
Angemeldet | 141.44.138.14
19
Scheibe, M., “Die Fördercharakteristik einer Zellenradschleuse unter
Berücksichtigung der Wechselwirkung von Silo und Austragorgan”, 1997,
Diss. (Ph.D. thesis) TU Bergakademie Freiberg.
Tomas, J. and S. Kleinschmidt, “Improvement of flowability of fine cohesive
powders by flow additives”, Chemical Engineering Technology, 2009, 32,
1470-1483.
Tomas, J., “Modellierung des instationären Auslaufverhaltens von kohäsiven
Schüttgütern aus Bunkern“, Chemische Technik, 1991, 43, 307-309.
Tomas, J., “Fundamentals of Cohesive Powder Consolidation and Flow”, Granular Matter, 2004, 6, 75-86.
Tomas, J., “Adhesion of ultrafine particles - a micromechanical approach”,
Chemical Engineering Science, 2007, 62, 1997-2010.
Angemeldet | 141.44.138.14

Hopper discharge rate

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