Micro- and Macromechanics of Hopper Discharge of Ultrafine Cohesive Powder

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1 INTERNATIONAL JOURNAL OF CHEMICAL REACTOR ENGINEERING Volume Article A44 Micro- and Macromechanics of Hopper Discharge of Ultrafine Cohesive Powder Jürgen Tomas Guido Kache Otto von Guericke University, Magdeburg, Polysius AG, Beckum, ISSN DOI: / A44 Copyright c 2012 De Gruyter. All rights reserved.

2 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 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.

3 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

4 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 = FG + FW + FT + FF (1) Dead weight as sole driving force (without vertical pressure onto the bridge): F G = ρ g b l dh (2) b B Published by De Gruyter, 2012

5 2 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Article A44 Wall force: F W = sin( θ + ϕ ) σ ϕ dh (3) W ' 1 2l cos( θ + W ) B 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: F T dv = ρ b b l dhb (4) dt Fluid drag force: F F dp = b l dhb (5) dh B 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

6 Tomas and Kache: Hopper Discharge of Ultrafine Cohesive Powder 3 Jenike s hopper design method (Jenike 1964), m = 0 wedge-shaped and m = 1 conical hopper): b min = ρ b, crit 2 ( m g + 1) sin2( ϕw + θ ) ( 1+ sinϕi ) sinϕst σ 0 [ 1 sinϕ sinϕ ( sinϕ sinϕ ) ( 2 ff 1) ] st i st i (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 ( sinϕst sinϕ 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) tanθ b dp dt b 2 ( m+ 1) tanθ dhb 1 ρ u b v b = g b r min (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- Published by De Gruyter, 2012

7 4 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Article A44 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 = , the classical threeterm model of Kaskas (1970) is preferred that includes the contributions of laminar (24/Re), transition range (4/Re 0.5 ) and turbulent (0.4) flow-around resistances: 24 4 c D = (8) Re Re 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): dp dh B 3 = Eu 4 B ρ f ( Re) d ST 1 ε u ε 2 r (9) u r d ST ρ f Re = (10) ε η 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

8 Tomas and Kache: Hopper Discharge of Ultrafine Cohesive Powder 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): 3 3 V s π d π d = = 3 V 6 l ε = (11) ( d a) 3 3 d ST 1 ε = 3 (12) a ε The maximum packing density of polydisperse particle bed is selected here to be 3 1 ε 0. or (1-ε) max = ( ) 95 max = Eu B 24 d = Re a ST 1 d + 2 a ST dst Re a dst a Re (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 Published by De Gruyter, 2012

9 6 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Article A44 ε 1 B c D 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, Eu lim =, obviously results in the settling of single particles that includes their flow-around and drag coefficient cd, 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: v st = b min b g 1 b b 2 ( m+ 1) tanθ ( m+ 1) tanθ dp dh B 1 ρ u b 2 r (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 v= 0 t dv g b = min 1 dt, (15) v v v b st st t= 0

10 Tomas and Kache: Hopper Discharge of Ultrafine Cohesive Powder 7 which results in the typical discharge velocity-time function of process dynamics: t v( t) = vst tanh (16) t76 t 76 = vst b g 1 b min = 1 b g 1 b min b min b g 1 b b 2( m+ 1)tanθ 1 + 2( m+ 1)tanθ dp dh B 1 ρ u 2 b r (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 ) = 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. () 3 = 0. st v( t99 = 3 t76 ) = v st tanh 995 v (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& ( t) = ρ, A v( t) (19) d b crit d 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 t h(t) = v(t) dt = vst t76 ln cosh (20) t= 0 t76 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- Published by De Gruyter, 2012

11 8 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Article A44 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): h hopper 2 hhopper td = + t76 ln exp (21) vst vst t76 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 1 exp (22) st vst t76 In this context, the discharge time-height function, Eq.(21), is extended by the filled hopper volume V hopper : V hopper 2 Vhopper td = + t76 ln exp (23) Ad vst Ad vst t76 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

12 Tomas and Kache: Hopper Discharge of Ultrafine Cohesive Powder 9 discharge velocity v st the last term in Eq.(23) can be simplified. For V /( A v t76 ) > 2 and consequently 1 exp( 4) = 0.98 follows: hopper d st 1 Vhopper td + t76 ln 2 (24) A v d st The first term is equivalent to the averaged residence time, with t V during steady-state discharge of a mass flow hopper. This mass flow pattern is equivalent to plug-flow and its uniform residence time distribution V hopper A v d st V = V& hopper st = t V (25) 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- Published by De Gruyter, 2012

13 10 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Article A44 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 f hopper interface a e f acceleration sensors m& vibrating hopper a e 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

14 Tomas and Kache: Hopper Discharge of Ultrafine Cohesive Powder 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 Silo diameter D in mm Hopper angle θ in deg Diameter D of vibrating hopper in mm Outlet diameter b outlet of vibrating hopper in mm 273 Annular gap widths b gap in mm Used silo volume V in m³ 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). Published by De Gruyter, 2012

15 12 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Article A conical outlet angle conical hopper wedge shape outlet / annular gap angle wedge shaped hopper 35 outlet width b min in mm ,05 0,1 0,15 0,2 0,25 0,3 peak oscillation velocity v peak 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) maximum hopper angle Θ max in degree 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 Vhopper 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. 1 st step annular gap: m& ρ A v (27) I = b, gap gap gap

16 Tomas and Kache: Hopper Discharge of Ultrafine Cohesive Powder 13 2 nd step outlet: m& ρ A v (28) II = b, outlet outlet outlet limiting condition of mass flow rates to avoid undesired consolidation within hopper V hopper : m & & (29) I m II v gap m& I v gap V hopper b gap Θ v outlet m& II b outlet 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 Published by De Gruyter, 2012

17 14 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Article A44 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: A gap 2 ( D bgap bgap = π ) (30) A π = (31) 4 2 outlet b outlet 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 v st,gap in m/s I II test rig A test rig B model (test rig A) model (test rig B) cross-sectional area of annular gap A gap in m² 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

18 Tomas and Kache: Hopper Discharge of Ultrafine Cohesive Powder 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. dm = V dt hopper d ( ρ Ψ) b dt = m I m II (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 ρb Vhopper = Ψmax (33) m& m& I II 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: t idle ρb Vhopper = Ψmax (34) m& II Published by De Gruyter, 2012

19 16 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Article A44 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 particle surface separation, µm A area, m² b given hopper outlet width, m b min minimum hopper outlet width to avoid bridging, m c D drag coefficient d particle size, µm D silo diameter, m d ST particle surface (Sauter) diameter, µm Eu Euler number F force, kn g gravitational acceleration, m/s²

20 Tomas and Kache: Hopper Discharge of Ultrafine Cohesive Powder 17 h height, m h 0 initial suspension height, m l length, m m hopper shape parameter m& powder mass flow rate, kg/s p fluid pressure Re particle Reynolds number t time, s u fluid velocity, m/s v discharge velocity (flow rate) of powder bridge, m/s V silo or hopper volume, m³ Greek letters ε η θ ρ σ c σ 1 σ 1 σ 0 τ φ i φ st φ W 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 Published by De Gruyter, 2012

21 18 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Article A44 max maximum min minimum op operation r relative st stationary T inertia V residence W wall x average of terminal value REFERENCES Brauer, H., Stoff-, Impuls- und Wärmetransport durch die Grenzfläche kugelförmiger Partikel, Chemie Ingenieur Technik, 1973, 45, Geldart, D., Types of Gas Fluidization, Powder Technology, 1973, 7, Haider, A. and O. Levenspiel, Drag coefficient and terminal settling velocity of spherical and nonspherical particles, Powder Technology, 1989, 58, 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, 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, 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, 8 th European Congress of Chemical Engineering (ECCE8), 2011, Berlin

22 Tomas and Kache: Hopper Discharge of Ultrafine Cohesive Powder 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, Tomas, J., Modellierung des instationären Auslaufverhaltens von kohäsiven Schüttgütern aus Bunkern, Chemische Technik, 1991, 43, Tomas, J., Fundamentals of Cohesive Powder Consolidation and Flow, Granular Matter, 2004, 6, Tomas, J., Adhesion of ultrafine particles - a micromechanical approach, Chemical Engineering Science, 2007, 62, Published by De Gruyter, 2012