10.1 Powder mechanics

Fluid and Particulate systems 424514 /2014 POWDER MECHANICS & POWDER FLOW TESTING 10 Ron Zevenhoven ÅA Thermal and Flow Engineering ron.zevenhoven@abo.fi 10.1 Powder mechanics RoNz 2/38

Types of flow of powders in silos [2] a. Mass flow b. Funnel flow c. Expanded flow d. Pipe e. Rathole f. Arching Differences : - Residence time variations - Wear of wall - Food / non-food RoNz 3/38 Stresses in a powder http://dietmar-schulze.de/spanne.html In solids, horizontal and vertical stresses are different, σ h /σ v typically ~ 0.3... 0.6 In fluids (non-flowing) σ h /σ v ~ 1 Fluids: surface tension, viscosity Powders: cohesion fluid powder RoNz 4/38

Stresses experienced by particles in powder [1] RoNz 5/38 Powder (flow) behaviour Powder will flow when a critical stress is overcome At low stress powder won t flow, and forms heaps. In a powder flow stress are not dependent on flow rate. Before flow, powders swell. Powders can consolidate, compact, decreasing porosity. Powders tend to form dense masses. Fluidisation is possible. Powders can segregate: particles with different shape, size, density,... will respond differently to forces and strains and as a result follow different paths. Source: K. Heiskanen, 2000 RoNz 6/38

Normal stress, shear stress & sliding [4] At contact point: normal force and shear force Negative normal force = tensile force For non-cohesive powders: normal force can only be compressive If shear force > ƒ normal force, with ƒ = coefficient of friction, then the powder granules will slide RoNz 7/38 http://www.sv.vt.edu/classes/esm4714/methods/eeg.html Stress components note: stress is result of strain Under a certain orientation, along the principal axes, the shear stresses go to zero. The stresses are then the principal stresses. With maximum stress σ 1, intermediate stress σ 2, minimum stress σ 3. Sum σ 1 + σ 2 + σ 3 = constant and irrespective of direction. Of interest here are primarily (only) σ 1 and σ 3, when an axis of symmetry exists then σ 3 = σ 2. RoNz 8/38

Transformation of stresses into major & minor principal stress xx and yy [4] eliminate α: 2 1 2 2 1 2 2 2 2 is a circle in σ, τ plot RoNz 9/38 Arching in silos (hoppers) [8] Important for silo/hopper design is the condition of collapse of the arch outlet size B design. The unconfined yield stress f c is the maximum normal stress under which a powder with a free, unstressed surface will yield RoNz 10/38

Yield locus of a Coulomb powder [1] = c +.tan τ c = cohesion ϕ = internal angle of friction Usually the Yield Locus is curved! 2 RoNz 11/38 Mohr circle + yield locus [4] shear stress < critical value, except at points C and D RoNz 12/38

Yield locus & Mohr circles for Coulomb powder A: no flow, B: flow [3] τ = c + σ tan (φ) φ 0 1 1 sin 2c cos 3 1 sin 3 1 sin 1 1 sin 1 sin o if c 0 1 sin 1 sin 3 φ internal angle of friction φ o effective internal angle of friction o Tensile strenght σ t σ 3 σ 1 RoNz 13/38 10.2 Powder flow testing, shear testing RoNz 14/38

http://www.freemantech.co.uk/images/stories/powdertesting/report_mohr_lg.gif Jenike shear tester (~1979) [1,5] Moves at 1-3 mm/min Force measured with for example a piezo-electric element RoNz 15/38 Principles of Jenike shear testing /1 [e.g. 5,6] Determines the shear strength of a particulate material under a certain load. Results in yield locus (Y.L.) for a certain consolidation. Y.L. gives minimum shear stress,, needed to initiate flow, as a function of the normal stress,, on the shear plane for steady state flow at a given bulk density. The consolidation can be quantified by a Mohr circle. Coulomb flow is assumed, i.e. = A + B Material sheared against itself angle of internal friction, effective angle of internal friction Material sheared against wall angle of wall friction RoNz 16/38

Dimensions of the Jenike shear cell [5] http://www.solidshandlingtech.com/ Weight for normal stress RoNz 17 Jenike shear cell testing [5] Experience: pre-shear ~ 0.08 powder density (kg/m 3 ) and multiples of that. Stress-strain curves for over- (1), critically (2) and under- (3) consolidated samples Stress-strain curve: pre-shear, weight reduction, shear RoNz 18/38

Yield loci and Mohr circles for shear test data [5,8] A family of yield loci f c σ max Yield locus showing valid shear points RoNz 19/38 Yield locus measurement [8] See next slide RoNz 20/38

Yield locus measurement [8] See previous slide RoNz 21/38 Typical data: free-flowing powders [8] Lucite = Perspex RoNz 22/38

Typical data: cohesive powders [8] Warren-Spring (WS) equation: Or, a linear fit: FF = flow function using parameters from the table RoNz 23/38 10.3 Silo design RoNz 24/38

Silo types, flow types Silo s have typical sizes of the order of 1000 m 3, say, 800 3500 m 3. Larger than 1000 m 3 : concrete, smaller than 650 m 3 : steel. Very large mammoth silo s can contain up to 100 000 m 3, with diameters up to 70 m, heights up to 30 m. Powder flow type depends on Internal friction of material Wall friction of material Half-angle of cone http://www.freemantech.co.uk/images/stories/ft4/ft4hopperdiag.jpg RoNz 25/38 Jenike shear testing & silo design /1 [e.g. 3,6] The shear test data yield locus Y.L. Mohr circle through consolidation point gives Mohr circle with minor and major consolidating stresses min and max, respectively. Mohr circle through 0 (i.e. σ 3 =0), tangential to Y.L., gives unconfined yield stress, f c, which is the normal stress needed to break and arch at a free surface (such as a hopper outlet). Flow Function, FF = max /f c gives first information : FF < 2 : extremely cohesive, 2 < FF < 4 : cohesive, 4 < FF < 10 : easy flowing, FF > 10 : free flowing. Combined with flow factor, ff, from design charts, a critical aperture size can be estimated using a flow/no flow analysis. RoNz 26/38

The Jenike Flow / No flow criterion [3,6,8] max / ff > f c,crit f c (Pa) & max (Pa) f c, crit max = 1/ff max Flow function, FF Flow No Flow max, crit max (Pa) RoNz 27/38 Data: low function and effective angle of friction for several common materials [8] Consolidation: σ n = 1.96, 4.90, 7.84 kpa RoNz 28/38

Flow function, ff, for a conical hopper [1] for given hopper half-angle ; angle of wall friction, w ; effective angle of friction for the powder, = 50 RoNz 29/38 Flow function, ff, for two hoppers [8] for given hopper half-angle ; angle of wall friction, w ; effective angle of friction for the powder, = 40 Square; length L > 3x width B RoNz 30/38

Jenike shear testing & silo design /2 [e.g. 6] Y.L. angle of internal friction,, and effective angle of wall friction,, Wall Y.L. angle of wall friction w. Mass flow criterion for conical hopper : << 45 - w i.e. = 45-1.2 w where = half-angle with respect to gravity. Minimum bottom opening d crit = f c,crit H( )/(g bulk ) using design graphs for H( ) Rule of thumb : d crit > 7 x particle size... For a conical hopper: H(α) = 2 + 0.0166 α RoNz 31/38 Function H( ) for hopper aperture versus hopper half-angle (conical and pyramidal hoppers) [1,8] Many empirical design diagrams like this are used! RoNz 32/38

10.4 Exercises 15 RoNz 33/38 Exercises 15 a. With a Jenike standard shear tester a powder with a particle size < 10 μm is being sheared, giving the tabelised (σ, τ) data for each preshear / shear test combination. See the three tables below (also next page) Produce the 3 yield loci for these tests, create the Mohr circles, determine the unconfined yiels stress c and the flow factor FF. Is this a cohesive powder? σ pre-shear τ pre-shear σ shear τ shear 2 1.8 1.5 1.62 2 1.97 1.5 1.67 2 1.87 1 1.35 2 1.94 1 1.39 2 1.92 1 1.43 2 1.87 1 1.45 2 1.85 1 1.47 2 1.8 1 1.46 average τ pre-shear : 1.88 σ pre-shear τ pre-shear σ shear τ shear 5 4.5 4 4.04 5 4.43 4 3.90 5 4.1 3 3.19 5 4.67 3 3.08 5 4.43 2 2.66 5 4.5 2 2.74 5 4.6 1 1.83 5 4.4 1 1.87 average τ pre-shear : 4.45 RoNz 34/38

Exercises 15 σ pre-shear τ pre-shear σ shear τ shear 9 8.3 7 6.9 9 8.2 7 7.2 9 8.1 5 5.6 9 8.6 5 5.5 9 7.9 3 4.1 9 8.6 3 4.0 9 7.9 1 2.40 9 8.7 1 2.34 average τ pre-shear 8.3 b. The powder discussed in question 15a is to be stored in a conical hopper under mass flow conditions. Shear tests with a sample of the wall material give an angle of wall friction φ wall =33. Using the Figures given here (for δ=φ i = 50 ), deterimine the angle α the flow function for this hopper the critical unconfined yield stress, c,crit the minimum bottom opening d crit Assume a powder bulk density ρ bulk =900 kg/m³. RoNz 35/38 T Exercises 15 RoNz 36/38

References [1] Iinoya, K., Gotoh, K., Higashitani, K. Powder technology handbook, Marcel Dekker, New York (1991) Chapters II.1, II.3, III.12, V. 4 [2] F.J.C. Rademacher Storage and handling of bulk powders (in Dutch), ProcesTechnologie Oct. 1992 36-41 & Nov. 1992, 27-32 [3] H. Rumpf Mechanische Verfahrenstechnik (in German) Carl Hanser Verlag, München/Wien (1975) Chapter 3.4 [4] A. Verruijt Soil mechanics (in Dutch) Delft Publishing Co., Delft (1983) [5] Standard shear cell testing technique (for particulate solids using the Jenike shear cell) Institute of Chemical Engineers, European Federation of Chemical Engineering, Rugby (UK) (1989) [6] Zevenhoven, C.A.P. Particle charging and granular bed filtration for high temperature application PhD thesis Delft Univ. of Technol., Delft (1992) Chapter 9.3 [7] L-S Fan, C Zhu Principles of gas solid flows, Cambridge Univ. Press (1998) Chapter 8 [8] C06: Crowe, C.T., ed., Multiphase Flow Handbook. CRC Press, Taylor & Francis (2006), Chapter 9 See also: D. Schulze Powders and bulk solids, Springer (2008) RoNz 37/38 End of course 424514 / year 2014 next course Jan/Feb 2016... RoNz 38/38