Fig Prof. Dr. J. Tomas, chair of Mechanical Process Engineering

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1 Fig Particle interactions, powder storage and flow 6. Dynamics of a flowing particle packing 6. Fundamentals of particle adhesion and adhesion forces 6.3 Mechanics of particle adhesion 6.4 Testing methods of particle adhesion 6.5 Flow properties of cohesive powders 6.6 Testing devices and techniques of powder flow properties 6.7 Applications in silo hopper design 6.8 Evaluation of residence time distribution of processes Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.

2 Fig. 6. Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.

3 Fig. 6.3 Survey of constitutive functions, processing and handling problems of cohesive powders Property, problems Large adhesion potential ) Large intensification of adhesion ) Poor flowablity ) Large compressibility ) Small permeability 3,4) Physical principle F N F N σ F H F H (F N ) F G Physical assessment of product quality Physical law Assessment characteristic Value Evaluation range C H H,sls Adhesion ( µm) - slightly adhesive = G πg a ρs d Weight d - 4 adhesive 4-8 very adhesive FH = κ FN + FH Contact consolidation coefficient κ.3.77 very soft..3 soft + F H by flattening >.77 extreme soft F F F H (F N ) ( ) h h W u σ c σ h b ρ ρ b b, ff c σ = σ c σ = + σ u = k f M,st h h Poor fluidisation 5,6) p = f ( u(d )) W b P n Flow function ff c < Compressibility index n Permeability k f in m/s < cohesive very cohesive non-flowing compressible very compressible non-permeable very low low Channelling Group C, non-fluidising Particle size d in µm < < <. < < <. < < < - - < ) Rumpf, H.: Die Wissenschaft des Agglomerierens. Chem.-Ing.-Technik, 46 (974) -. ) Tomas, J.: Product Design of Cohesive Powders - Mechanical Properties, Compression and Flow Behavior. Chem. Engng. & Techn., 7 (4) ) Förster, W.: Bodenmechanik - Mechanische Eigenschaften der Lockergesteine, 4. Lehrbrief, Bergakademie Freiberg ) Terzaghi, K., Peck, R. B., Mesri, G.: Soil mechanics in engineering practice, Wiley, New York ) Geldart, D.: Types of Gas Fluidization, Powder Techn. 7 (973) ) Molerus, O.: Fluid-Feststoff-Strömungen, Springer, Heidelberg 98. Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.3

4 Fig. 6.4 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.4

5 Fig. 6.5 Interactions of Polar Molecule Pair interaction pair potential U in - J repulsion -5,,,,3,4,5-5 - attraction total potential U total force F bond energy U B repulsion potential U ab repulsion force F ab a U= a F= a Fmax potential force F in - N Interaction pair potential due to MIE (93) and e.g. the LENNARD-JONES potential: U - A B = n + m a a integer exponents n < m () U 4 U Pot. equilibrium separ.: a Bond energy: U B U= B = A m n m n A = n m a F= Maximum attraction force: d U df = = : F Separation ratios: da da a U = B a = = (3) equilibrium separation: a (5) potential ratio: U U max a = < = m m n < a F F = m ( m + ) max a n a n ( n + ) U= U= B = an a F= a U a max F= = 6 m B n A m n () m n < (6) m m n n A = n m + + a (7) F a a a U= Fmax Strain: εu= = = < εf= = < εf = max a a a (9) Modulus of elasticity: attraction potential U an atomic centre separation a in nm F= m n F= ( U ) d U A B E = = ( m n) n n+ 3 = n m 3 () a da a a F= a F= F= attraction force F an F= - (8) (4) Pull-off strength: σ Z,max E = n m + + m + n+ m n () Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.5

6 Fig. 6.6 Interaction Forces and Potentials between Smooth and Stiff Model Bodies Partner Dipol moment and dispersion Electrostatic (COULOMB) (VAN-DER-WAALS) Molecule-molecule a Α U VdW = - Sphere - sphere Point charge sphere-sphere q = ε ε r U el /d d a d = d d d + d d E VdW = F VdW = - - C H d 4 a C H d 4 a² Molecule - plate Sphere - plate Conductor Non-conductor ll a a E ρ VdW = - n, a a 6 6 A F VdW = - a 7 d M d A S C H d a C E VdW = - H l d 4 a 3/ C F VdW = - H l d 6 5/ a Q = π d ε ε r E Q = A S ε ε r E C F VdW = - H d π ρ n, A a² π a π d E U VdW = - el = ε ε r U el d ln E d el = q q a ε 6 a³ ε r π ρ F VdW = - n, A π d π d F C = ε ε r U el F C = q q a 4 a ε ε r Parallel chain Cylinder - cylinder Conductor Non-conductor molecules d d Q = π d l ε ε r E 3 π U VdW = - A l 8 d M a 5 5 π A l F VdW = - 8 d M a 6 HAMAKER constant = f(a): C H = π ρ n, ρ n, A l l d = d d d + d Plate - plate Conductor Non-conductor a Crossed cylinders d A S a a E VdW = F VdW = d ² E VdW = F VdW = - C H A S π a² - C H A S 6 π a 3 - C H d d a - C H d d a² Q = A S ε ε r E A E el = S ε ε r U el a A F C = S ε ε r U el a l d E el = q q a ε ε r l d F C = q q ε ε r A S E el = q q a ε ε r A S F C = q q ε ε r n q = Q/A S = Q e A ε ε (+ r,s - ) E surface charge density E electric field strength S ε r,s + U el electrostatic potential ρ n = ρ N A /M number density F = - du/da potential (counter) force e =.6-9 A s electronic charge z ion valency ε = A s/(v m) permittivity of vacuum permittivity of interstitial medium ε r U C = z z e² 4π ε a z z e² F C = 4π ε a² E el = F C = E el = F C = l d ε ε r U el a l d ε ε r U el a π q q d d ε ε r (d + d + a) π q q d d ε ε r (d + d + a) ISRAELACH VILI, J.: Intermolekular & Surface Forces, Academic Press London 99, p.77 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.6

7 Fig. 6.7 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.7

8 Adhesion Forces between Stiff Solid Particles Models according to Rumpf et al. (974): a) Smooth sphere - smooth plate b) Rough sphere - smooth plate Fig. 6.8 a α d adhesion force F H in N liquid bridge van der Waals, dry conductor van der Waals, wet non-conductor weight of sphere h r a d adhesion force F H in N µm µm liquid bridge, d = µm d = µm µm, α =,5 van der Waals conductor, d = µm non-conductor, d = µm - 3 particle size d in µm 3 roughness height. h r in nm acc. to H. Schubert (979): adhesion force F H in nn 4 - α = liquid bridge rel. humidity 5% van der Waals a =.4 nm conductor non-conductor adhesion force F H in N C H h r d / h F H,VdW =. r 6 a +. ( + h r / a ) h r = nm nm 5 nm µm nm nm -4-3 particle separation a in nm particle size d in µm a =.4 nm molecular force equilibrium separation σ lg = N/m surface tension α = bridge angle θ = wetting angle C H = 9. - J Hamaker constant acc. to Lifschitz q max = 6-9 As/µm surface charge density U =.5 V contact potential C H,sls = ( C H,ss - C H,ll ) Hamaker constant particle-water-particle Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.8

9 d/ Fig Bond types a) Pendular state (liquid bridges) for a real packing: Moisture Bonding in a Particle Packing b) Funicular state (bridges + filled pores) c) Capillary state (filled pores) for cubic packing of monodisperse spheres:. Liquid bridge at direct contact (a = a F= ) of two equal-sized spheres α σ lg F H h F s R R R' F s d/ σ lg Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.9

10 Fig. 6. Moisture Bonding in a Particle Packing capillary pressure p K. Capillary pressure hysteresis of a particle packing dewatering p Ke adsorption capillary condensation moisten saturation X WC water content X W X WS. Sorption isotherme of capillary-porous particles and packings water content X W Adsorption Desorption X WK capillary condensation multimolecular layers monolayer ϕ K relative partial pressure ϕ = pi/psi Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.

11 Fig. 6. Crystallisation Bridge between KCl 99 Particles d = - 6 µm Bulk caking and hardening in store house: Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.

12 Fig. 6. Stress Transmission at Time Consolidation (Caking) σ ct (t) A tot A sf (t) σ Dsf A tot dσct (t) dt = σ Dsf dasf (t) dt () dσct(t) dt = σ Dsf dv V sf tot (t) dt () σ ct (t) = σ Dsf ( ε) ρ ρ s sf t L dm m s sf (t) dt dt (3) Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.

13 Fig. 6.3 Particle Contact Deformation in Normal Direction without Adhesion r rk,el F N F N r K,el r << h K,el r K,pl r F N F N r K,pl r << h K,pl contact normal force F N 3π p f E* r h K,f = ( ) yielding loading W D = F R (h K ) dh K k N = df N dh K unloading particle centre approach h K stiffness k N = force response F R = deformation work W D = elastic E* d h K,el 3 E* 3 d h K,el 5 E* d 5 h K,el plastic and viscoplastic π r p f π r p f h K,pl π r η K h K,vis π r p f (h K,pl - h K,f ) π r η K h K,vis t material data: E* effective modulus of elasticity, p f micro-yield strength, η Κ contact viscosity Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.3

14 Fig. 6.4 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.4

15 Testing the Adhesion Force between Particle and Surface Fig. 6.5 a) Spring balance method b) Centrifugal method Pressing Detachment F N F H F C F H c) Vibration method d) Impact separation method e) Hydrodynamic method u H. Masuda and K. Gotoh, Adhesive Force of a Single Particle, pp.4, in K. Gotoh, M. Masuda, K. Higashitani, Powder Technology Handbook, Marcel Dekker, New York 997 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.5

16 Fig. 6.6 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.6

17 Fig. 6.7 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.7

18 Fig. 6.8 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.8

19 Fig. 6.9 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.9

20 Fig. 6. Biaxial Stress States of Sheared Particle Packing () shear and dilatancy dv > shear stress τ τ c yield locus ϕ i angle of internal friction normal stress σ τ cohesion τ c σ angle of dilatancy ν (+) h shear stress τ τ c yield locus τ c ϕ i uniaxial pressure σ c σ c σ c normal stress σ uniaxial tension σ Z σ Z σ Z σ Z shear stress τ τ c yield locus ϕ i σ Z σ Z tension isostatic σ Z σ Z σ c normal stress σ Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.

21 Fig. 6. Biaxial Stress States of Sheared Particle Packing () stationary shear dv = isostatic tensile strength σ σ shear stress τ yield locus ϕ st stationary yield locus σ R,st stationary angle of internal friction τ σ σ σ σ normal stress σ σ M,st σ σ σ no deformation (3) shear and compression dv < τ σ h shear stess τ yield locus ϕ i ϕ i σ R,st ϕ i ϕ i ν (-) angle of dilatancy consolidation locus σ σ σ Z normal stress σ σ M,st σ iso isostatic pressure, compression dv < σ iso σ iso σ iso σ iso Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.

22 Fig. 6. Biaxial Stress States of Sheared Particle Packing a) The three flow parameters shear stress τ Stationary Yield Locus Yield Locus End point ϕ i ϕ i ϕ st σ angle of internal friction, stationary angle of internal friction, isostatic tensile strength of unconsolidated packing; and σ M,st centre stress for steady-state flow σ ϕ st normals stress σ σ M,st b) Stress states Stationary Yield Locus shear stress τ τ c Yield Locus σ R,st σ c σ M,st normal stress σ σ σ Z Z σ σ VM σ σ σ c σ Z σ Z σ iso major principal stress, minor principal stress, uniaxial compressive strength, uniaxial tensile strength, isostatic tensile strength, isostatic pressure; Consolidation Locus σ VR ϕ i σ σ iso c) Stress states at Mohr circle of steady-state flow: shear stress τ σ Z σ Stationary Yield Locus Yield Locus ϕ i End point ϕ st σ R,st ϕ st τ st σ σ st σ M,st normal stress σ Stationary Yield Locus: σ R,st = sinϕ st. (σ M,st + σ ) Tangential point: τ st = cosϕ st. σ R,st σ st = σ M,st - sinϕ st. σ R,st Yield Locus: τ = tanϕ i. (σ + σ Ζ ) Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.

23 Fig. 6.3 Yield Loci and Powder Flow Parameters for: a) a dry, cohesion-less or free flowing particulate solid τ E ϕ i ϕ i = ϕ st σ b) a general case of moist or fine cohesive powder τ ρ b = const. E A ϕ i τ c ϕ st σ Z(3) -σ Z() σ c σ σ c) a wet-mass viscoplastic powder without Coulomb friction τ. τ = f ( γ ) ϕ i = A preshear point E end point. γ shear rate gradient ρ b bulk density σ major principal stress σ c uniaxial compressive strength σ z() uniaxial tensile strength σ z(3) isostatic tensile strength τ c cohesion ϕ st stationary angle of internal friction angle of internal friction ϕ i σ Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.3

24 Fig. 6.4 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.4

25 Fig. 6.5 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.5

26 Fig. 6.6 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.6

27 Fig. 6.7 Incipient Yield and Steady-State Flow F N shear preshear F N F S F S s s shear force F S preshear plastic yielding dv= σ pre shear dv> σ<σ pre σ pre shear stress τ = F S / A steady-state flow incipient yielding instantaneous yield locus displacement s σ normal stress σ = F N / A σ pre Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.7

28 Fig. 6.8 Instantaneous, Stationary, Time Yield Locus and Wall Yield Locus F N preshear time consolidation t >> F N F N shear F N s wall shear FS FS F S F S FN s t >> s shear force F S preshear shear σ<σ pre σ pre σ pre σ>σ pre shear stress τ = F S / A τ c time yield locus ϕ st ϕ it steady-state flow incipient yielding ϕ i end point yield locus stationary yield locus ϕ W time t (or displacement s = v S. t) wall yield locus displacement s σ Z σ σ σ c σ M,st σ ct normal stress σ = F N / A Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.8

29 uniaxial compressive strength σ c in kpa -5. ff c σ c (σ ) t = σ ct (σ ) t = 4 h ff c < hardened non flowing σ Fig. 6.9 Consolidation Function of Titania particle size d S = nm, moisture X w =.4%, temperature = C sin ϕ = i + ( + κ) + + ( + κ) ( + κ) tan ϕi ( + κ) tan σ c consolidation stress σ in kpa tan ϕ i i tan ϕ ϕ i sin ϕ i < ff c < very cohesive < ff c < 4 cohesive 4 < ff c < easy flowing free flowing ff c = σ /σ c Flowability assessment and contact consolidation coefficient κ(ϕ i = 3 ) flow function κ-values ϕ st in deg evaluation examples ff c free flowing dry fine sand easy flowing moist fine sand cohesive dry powder very cohesive moist powder < - non flowing, hardened (ff ct ) moist powder hydrated cement Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.9

30 Fig. 6.3 Isentropic Powder Compression Adiabatic gas compression: dv dp = κ ad V p Isentropic powder compression: ρ b dρ ρ b = n σ M,st ρb, b M,st σ dσ M,st + σ () () bulk density ρ b ρ b, n = ideal gas compressibility index < n < compressible n = incompressible σ M,st σ ρ b = ρ b, ( + ) n isostatic tensile strength -σ centre stress during consolidation or steady-state flow σ M,st Compressibility index of powders, semi-empirical estimation for σ = kpa index n evaluation examples flowability. incompressible gravel free flowing..5 low compressibility fine sand.5 -. compressible dry powder cohesive. - very compressible moist powder very cohesive Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.3

31 Fig. 6.3 Compliant and Stiff Particle Contact and Powder Behaviour a) particle contact deformation b) particle adhesion force F N -F H stiff compliant displacement h K adhesion force F H compliant stiff normal force F N shear stress τ c) powder yield loci d) consolidation functions σ cohesive YL YL SYL free flowing SYL normal stress σ uniaxial compressive/ tensile strength σ c, σ Z compliant cohesive stiff, free flowing consolidation stress σ e) powder constitutive models radius stress σ R YL σ cohesive YL SYL CL SYL CL free flowing average pressure σ Μ σ iso f) compression function bulk density ρ b ρ b, σ compliant compressible stiff, incompressible consolidation stress σ Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.3

32 Fig. 6.3 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.3

33 Fig Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.33

34 Fig Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.34

35 Fig Widely Spread Residence Time Distribution Q 3 (t V ) t v Storage Time too Large Q 3 (t V ) Storage Time too small: Time Consolidation Problems t v Q 3 (t V ) t v Deaeration Problems Inflammation or Explosion Hazards Deterioration Problems Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.35

36 Fig Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.36

Fig. 6.37 Solution for Silo Plant Design () Economic Goals Marketing Cost-Benefit-Analysis Investment Tasks +?

Testing of Bulk Materials F s F N Particle Size Distribution µm... cm? Moisture Storage Time Chem.-Min. Composition %o...%? h.

37 Fig Solution for Silo Plant Design () Economic Goals Marketing Cost-Benefit-Analysis Investment Tasks +? () Layout Flow Sheets Logistics (3) Storage Capacity Area Requirements Selection of Main Dimensions (4) Flow Behaviour Testing of Bulk Materials F s F N Particle Size Distribution µm... cm? Moisture Storage Time Chem.-Min. Composition %o...%? h... d? %? Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.37

38 Fig (5) Design, Shaft and Hopper Dimension???? (6) Handling Equipment Selection Feeder (Filling) Discharge Aids Gate Chute Feeder (Discharging) Conveyor (7) Apparatus Design and Adaption Mass Flow Rate Dosage Power (Consumption Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.38

39 Fig (8) Design of Dust Collection Level Control Devices Safety Instrumentation (9) Structural Design Silo Pressures Wall Thickness, Steel Reinforcement Plant and Building () Design and Construction, considering Environmental Protection Manpower and Social Services Manufacturing Environmental Protection Access and Cleaning Repair and Maintenance Safety Measurement and Control Instrumentation... etc Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.39

40 Fig. 6.4 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.4

41 Fig. 6.4 Bounds between Mass and Core Flow axisymmetric Flow (conical hopper) angle of wall friction ϕ w in deg Mass Flow Core Flow effective angle of internal friction ϕ e = hopper angle versus vertical Θ in deg Θ 8 - arccos - sin ϕ e - ϕ W - arc sin sin ϕ W sin ϕ e sin ϕ e select Θ:= Θ 3 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.4

42 Fig. 6.4 Bounds between Mass and Core Flow Plane Flow (wedge-shaped hopper) angle of wall friction ϕ w in deg Mass Flow Core Flow effective angle of internal friction ϕ e = hopper angle versus vertical Θ in deg Θ 6,5 + arc tan 5 -ϕ e 7,73 5,7 ϕ W - 4,3 +,3 exp(,6 ϕe ) with ϕ W < ϕ 3 and Θ e 6 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.4

43 b min Fig Conical Hopper (axisymmetric stress field) Cone Pyramid Θ max Θ max Θ wall b min shape factor m = max [ 3a ] - Wedge-shaped Hopper (plane stress field) vertical front walls Θ wall tanθ = arctan D Θ max l min > 3 b min b min l min >3 b min b min shape factor m = Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.43

44 Fig inclined front walls B L Θ max Θ max,5 b min 3 b min b min l min > 6 b min,5 b min Θ wall = arc tan Θ tan max [( B b) or ( L l) ] ( L l) + ( B b) [ 3b] Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.44

45 Fig Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.45

46 Fig Ascertainment of Approximated Flow Factor (angle of wall friction ϕ W = - 3 ) flow factor ff,5 conical hopper wedge-shaped hopper effective angle of internal friction ϕ e in deg Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.46

47 Fig Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.47

48 Fig Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.48

49 Calculation of Silo Pressures according to Slice-Element Method Force Balance F = Fig ρ b g dy Shaft (Filling F): H H Tr p v p p h p h W p v + dp v p v + dp v Θ p n p W ρ b g dy pw p v p n dy p W dy Θ da y H * y dp dy p p p v h w v with + λ = ρ = λ b F F H g 63 p = tanϕ tan ϕ v w λ F w U A F p v p v ρ H 63 exp A = U λ tan ϕ w b H H g = 63 Hopper: dpv + p dy dpv dy v k da dy pv k y + ρ A b g = ( tanθ + tanϕ ) with ρb g H Tr HTr H * HTr H * pv = k H Tr H Tr A with HTr =, pn = k pv U tanθ = k tanϕ tanθ H * and and w ( m + ) k + ( m + ) y = H Tr w = U A k + ρ + b p g = p v p v H w = tanϕ w p tanθ H * H Tr Tr v = tanϕ λ for w y k = p H v Tr Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.49

50 Fig. 6.5 Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.5

51 Fig. 6.5 Evaluation of Residence Time Distribution of Processes page. Batch process a) Cummulative residence time distribution F(τ) b) Frequency distribution f(τ) for τ<τ F( τ) = Θ( τ) = for τ τ F(τ) τ R τ. Continuous processes with ideal residence time distribution Block flow chart: tracer f ( τ) = m * dm dτ * R R * * m m ( τ) F( τ) = m * ( τ) * m m τ τ tracer m for τ τ f ( τ) = F( τ) = δ( τ) = for τ=τ b) ideal continuous mixer c) cascade of ideal continuous mixers f(τ) τ R a) ideal plug flow channel (piston flow, ideal replacement) τ R R [ [ [ [ 3. st and nd moment of residence time distribution a) mean residence time (complete initial moment) τ m = m m Fill = M,3 = N τ f ( τ) dτ c s, i= τ m,i ( c c ) s,i s,i b) variance of residence time distribution ( nd central moment) σ ( τ) = M,3 = c s N ( τ τm ) f ( τ) dτ = ( τ τm ) dc s ( τm,i τm ) ( cs,i cs,i ) c s, c s, c s, i= Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.5

52 Fig. 6.5 Evaluation of Residence Time Distribution of Processes page 4. Normalized frequency distribution of residence time τ m. f(τ/τ m ) for cascades of ideal mixers with various stage numbers n at constant total mean residence time τ m = n. τ m,n = const. τ m f (τ/τ m ),5,,5 n n= 6 3 n τ m f ( τ) = (n )! with mean residence time of one stage (unit): τ m,n τ τ m,n Vn V τm = = = V n V n n τ exp τ m,n,5,,5,,5 τ/τ m 5. Cummulative residence time distribution F(τ/τ m ) for cascades of ideal mixers with various stage numbers n F(τ/τ m ),8,6,4, n= 3 6 n n F( τ) = i = (i )! n = n τ τ m,n i ideal mixer τ exp τ ideal replacement m,n 3 4 τ/τ m 6. Flow channel (pipe) with axial redispersion v' x v τ m = x v v x Bo = D P Bodenstein number of particles Fig_MPE_6 VO Mechanical Process Engineering - Particle Technology Particle Storage and Transport Prof. Dr. J. Tomas.5.3 Figure 6.5 D P Λ Bo Bo v' axial dispersion (diffusion) coefficient of particles ideal mixer ideal replacement

THE MECHANICS OF DRY, COHESIVE POWDERS 1

THE MECHANICS OF DRY, COHESIVE POWDERS Jürgen Tomas Mechanical Process Engineering, The Otto-von-Guericke-University Magdeburg Universitätsplatz, D 39 6 Magdeburg, Germany Phone: ++49 39 67 8 783, Fax: