10.1 Powder mechanics - Åbo Akademi · 2018-05-08 · Powder (flow) behaviour Powder will flow when a critical stress is overcome At low stress powder won’t flow, and forms heaps

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10Fluid and Particulate Systems 424521 /2018

POWDER MECHANICS & POWDER FLOW TESTING(flow / no flow)

Ron ZevenhovenÅA Thermal and Flow Engineering

[email protected]

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10.1 Powder mechanics

Åbo Akademi University - Värme- och Strömningsteknik Biskopsgatan 8, FI-20500 Åbo / Turku Finland

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Types of flow of powders in silos [2]

a. Mass flow b. Funnel flowc. Expanded flow d. “Pipe”e. Rathole f. Arching

Differences :- Residence time

variations- Wear of wall- Food / non-food


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Stresses in a powder

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fluid powder

In solids, horizontal and verticalstresses are different,σh/σv typically ~ 0.3 ... 0.6

In fluids (non-flowing) σh/σv ~ 1

Fluids: surface tension, viscosity

Powders: cohesion


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SeeJanssen Equation(1895)

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Stresses experienced by particles in powder, stresses in a silo [1,9]


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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, etc. willrespond differently to forces and strains and as a result follow different paths.

No surface tension: particles can enter through small holes

Source: K. Heiskanen, 2000


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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.

ƒ = tan (φ), with angle of friction φ


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Stress componentsnote: stress is result of strain

Under a certain orientation, along the principal axes, the shearstresses go to zero. The stresses are then the principal stresses.With maximum stress σ1, intermediate stress σ2, minimum stress σ3.Sum σ1+ σ2+ σ3 = constant and irrespective of direction.Of interest here are primarily (only) σ1 and σ3, when an axis of symmetry exists then σ3 = σ2 .

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Transformation of stresses into major & minor principal stresses

xx and yy [4]

eliminate α: 2

2122

21

22

is a circle in σ, τ plot


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Yield locus of a Coulomb powder [1]

= c + ∙tan

2

τc = cohesion

ϕ = internal angleof friction

Usually the Yield Locus

is curved!


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Arching in silos (hoppers) [8]

Important for silo/hopper design is the conditionof collapse of the arch outlet size B design.

The unconfined yield stress fc is the maximum normal stress under which a powder with a free, unstressed surface will yield


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Yield locus & Mohr circles for Coulomb powder A: no flow, B: flow [3]

τ = c + σꞏtan (φ)

σ3 σ1

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φ internal angleof friction

φo effective internalangle of friction

Tensile strenght σt


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shear stress < critical value, except at point *

*

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Multiple yield loci [9]


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δ = effective angle of internal frictionJYL = Jenike yield locus

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10.2 Powder flow testing, sheartesting


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Jenike shear tester (~1979) [1,5]

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Force measured with for examplea piezo-electric element

Moves at 1-3 mm/min


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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 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


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Dimensions of the Jenike shear cell [5]

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Weight for normal stress


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Jenike shear cell testing [5]

Stress-strain curves for over- (1),critically (2) and under- (3)

consolidated samples

Stress-strain curve: pre-shear, weight reduction, shear

Experience: pre-shear (kPa) ~ 0.08×√powder density (kg/m3)and multiples of that.


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one test in two steps

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Yield loci and Mohr circles for shear test data [5,8]

A family of yield loci Yield locus showing validshear points

fc σmax


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Typical data: free-flowing powders [8]


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Lucite =Perspex

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Typical data: cohesive powders [8]


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Warren-Spring (WS) equation:

Or, a linear fit:

using parameters from the tableFF = flow function

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10.3 Silo design


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Silo types, flow types Silo’s have typical sizes of the order of 1000 m3, say,

800 – 3500 m3. Larger than 1000 m3: concrete, smaller than 650 m3: steel.

Very large ”mammoth”silo’scan contain up to 100 000 m3, with diameters up to 30 m, heights up to 70 m.

Powder flow type depends on Internal friction of material Wall friction of material Half-angle of cone

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Shear test data yield locus Y.L. Mohr circle through consolidation point gives Mohr circle

with minor and major consolidating stresses min and max. Mohr circle through 0 (i.e. σ3 = 0), tangential to Y.L., gives

unconfined yield stress, fc, which is the normal stress needed to break and arch at a free surface (such as a hopper outlet).

Flow Function, FF = max / fc 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.

Jenike shear testing & silo design /1 [e.g. 3,6]


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The Jenike Flow / No flow criterion [3,6,8]

fc (Pa)

max (Pa)

Flow function, FF’max (Pa)

&

’max = 1/ff max

max, crit

fc, crit

Flow

No Flow

max / ff > fc,crit


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Data: flow function and effectiveangle of friction for common

materials [8]


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Consolidation: σn = 1.96, 4.90, 7.84 kPa

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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°


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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°


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Square; length L > 3x width B

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Jenike shear testing & silo design /2 [e.g. 6,9]

Y.L. angle of internal friction, , and effective angle of friction, ,

Wall Y.L. angle of wall friction w. Mass flow criterion for conical hopper :

<< 45° - w i.e. = 45° - 1.2w

where = half-angle with respect to gravity. Minimum bottom opening dcrit = fc,crit H()/(g bulk)

using design graphs for H() “Rule of thumb” : dcrit > 7 x particle size...... For a conical hopper: H(α) = 2 + α/60 For a symmetrical slot outlet: H(α) = 1 + α/180, L > 3W


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Function H() for hopper aperture versus hopper half-angle

(conical and pyramidal hoppers) [1,8]

Manyempirical

designdiagramslike this

are used !


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Design charts [9]


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δ = effective angle of internal friction

δw = angle of wall friction

L > 3W

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Design charts [9]


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δ = effective angle of internal friction

δw = angle of wall friction

d

α

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10.4 Exercises 15


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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 pre-shear / 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?

σpre-shear

(kPa)τpre-shear

(kPa)σshear

(kPa)τshear

(kPa)

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 τpre-shear :

1.88

σpre-shear

(kPa)τpre-shear

(kPa)σshear

(kPa)τshear

(kPa)

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 τpre-shear :

4.45


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Exercises 15

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 φwall =33.Using the Figures given here (for δ=φi = 50), deterimine

the angle α

the flow function for this hopper the critical unconfined yield stress, c,crit

the minimum bottom opening dcrit

Assume a powder bulk density ρbulk= 900 kg/m³.

σpre-shear

(kPa)τpre-shear

(kPa)σshear

(kPa)τshear

(kPa)

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 τpre-shear

8.3


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Exercises 15T


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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 [9] George G. Chase, SOLIDS NOTES 10, The University of Akron OH, USA

available at: https://www.inti.gob.ar/cirsoc/pdf/silos/SolidsNotes10HopperDesign.pdf See also: D. Schulze “Powders and bulk solids”, Springer (2008)


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Documents

10.1 Powder mechanics - Åbo Akademi · 2018-05-08 · Powder (flow) behaviour Powder will flow when a critical stress is overcome At low stress powder won’t flow, and forms heaps