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Powder Flow: Theory, Characterisation and Application
Edited by
Massih Pasha
Massih Pasha
The Chemours Company, USA
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Hardback ISBN:
978-1-78801-224-9
PDF ISBN:
978-1-78801-610-0
EPUB ISBN:
978-1-78801-847-0
Special Collection:
2019 ebook collection
No. of Pages:
222
Published online:
12 Jul 2019
Published in print:
16 Jul 2019
Chapter Contents
Book Chapter
Nomenclature
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Published:12 Jul 2019
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Special Collection: 2019 ebook collection
Page range:
P007 - P012
Citation
Powder Flow: Theory, Characterisation and Application, ed. A. Hassanpour, C. Hare, and M. Pasha, The Royal Society of Chemistry, 2019, pp. P007-P012.
Download citation file:
Latin Characters
a | Inter-molecular distance | m |
aw | Adhesion | Pa |
aξ | Lack of velocity fluctuations | — |
A | Hamaker constant | J |
Ab | Cross-sectional area of the powder bed | m2 |
Aeff | Normalized effective projected area | m2 |
AI | Arching index | — |
Ap | Cross-sectional area of plate | m2 |
Aproj | Projected area of indent | m2 |
AR | Aspect ratio | — |
ARST | Cross-sectional area of Schulze ring shear cell | m2 |
ATD | Average time difference | s |
AU | Cross-sectional area of uniaxial compaction tester | m2 |
AJT | Cross-sectional area of Jenike cell | m2 |
b | Body force | N |
B | Distance of barycentre of mass from axis of rotation | m |
BIM | Ball indentation method | — |
Bo | Bond number | — |
c | Cohesion | Pa |
C | Constraint factor | — |
CBeverloo | Flowability index | — |
CI | Carr index | — |
Co | Compressibility | — |
Coh | Cohesion number | — |
d | Particle size | m |
d43 | Volume moment mean diameter | m |
db | Indenter diameter | m |
dc | Diameter of contact | m |
dFe | Feret diameter | m |
dFe⊥ | Feret diameter measured in perpendicular direction | m |
dv | Particle equivalent volume sphere diameter | m |
D | Aperture diameter | m |
Dheap | Diameter of powder heap | m |
e | Rod diameter | m |
eeff | Effective coefficient of restitution | — |
E | Young modulus | Pa |
Ef | Flow energy | J |
f | Inter-particle force | N |
fc | Unconfined yield stress | Pa |
ff | Flow-factor | — |
F | Force | N |
Fc | Contact force | N |
FD | Fractal dimensions | — |
Fg | Gravitational force | N |
F | Contact force exerted on i by j | N |
Fn | Normal load | N |
Fnc | Non-contact force | N |
Fr | Frounde number | — |
Fs | Macroscopic shear force | N |
Ft | Translational force | N |
g | Gravitational acceleration | m s−2 |
g0 | Radial distribution function | — |
Gn | Solid fraction dependent functions in rapid flow regime | — |
h | Height | m |
hB | Burke constant | — |
hc,max | Indentation unloading intercept with depth axis | m |
hd | Indentation depth | m |
hf | Final indentation depth | m |
hsep | Separation distance | m |
H | Indentation hardness | Pa |
HR | Haussner ratio | — |
i | Element under consideration | — |
I | Moment of inertia/inertial number | kg m2 |
I0 | Constant for μ(I) rheology | — |
IA, IB | Modified Bessel functions of the first kind | — |
k | Permeability | m2 |
kA | Fraction of collisional dissipation contribution to shear viscosity | — |
kB | Fraction of frictional contribution to shear viscosity | — |
kc | Carman constant | — |
kj | Constant for Johanson hang-up indiciser | ft |
Ka | Active Rankine coefficient | — |
Kp | Passive Rankine coefficient | — |
l | Rod length | m |
lij | Vector from mass centre of particle i to mass centre of particle j | m |
ℓp | Unit vector along the major axis | — |
L | Drum length | m |
Lp | Perimeter length | m |
Ly | Dimension of domain in y-direction | m |
m | Mass | kg |
M | Contact torque | N m |
n | Number of particles | — |
N | Macroscopic normal force | N |
Nc | Applied force at failure | N |
O | Order parameter | — |
p | Particle under consideration | — |
P | Pressure | Pa |
Pa | Active pressure | Pa |
Pp | Passive pressure | Pa |
PFI | Powder flow index | — |
PR | Packing ratio | — |
q | Energy flux | N m−1 s−1 |
Qij | Symmetric traceless order tensor | Pa |
r | Asperity size | m |
rcol | Radius of meniscus curvature | m |
rro | Surface roughness length | m |
R | Radius | m |
R1 | Minor radius of the lid cross-sectional annular area | m |
R2 | Major radius of the lid cross-sectional annular area | m |
Rgas | Ideal gas constant | kg m s−2 K−1 |
RH | Relative humidity | % |
RI | Ratholing index | — |
s | Particle surface area | m2 |
S | Stiffness | N m−1 |
Seff | Effective projected surface area of a particle | m2 |
Ssph | Effective projected surface area of the sphere of the same volume to the particle | m2 |
STD | Standard deviation | — |
t | Time | s |
T | Torque | N m |
Tabs | Temperature | K |
Tgran | Granular temperature | m2 s−2 |
U | Velocity at boundary | m s−1 |
Ucons | Consolidation velocity | m s−1 |
Umf | Minimum fluidisation velocity | m s−1 |
Uro | Rain-off velocity | m s−1 |
Ut | Terminal velocity | m s−1 |
v | Velocity | m s−1 |
vax | Axial velocity | m s−1 |
〈vp,ave〉 | Average particle velocity | m s−1 |
vp,fl | Fluctuating velocity | m s−1 |
vp,i | Velocity of particle i | m s−1 |
V | Particle volume | m3 |
Vliq | Volume of liquid per contact | m3 |
Vm | Molar volume | m3 |
w | Weight | N |
W | Mass flow rate | kg s−1 |
x | Distance between crossbeam holders | m |
y | Parameter of yield locus curvature | — |
Y | Yield stress | Pa |
Z | Height | m |
a | Inter-molecular distance | m |
aw | Adhesion | Pa |
aξ | Lack of velocity fluctuations | — |
A | Hamaker constant | J |
Ab | Cross-sectional area of the powder bed | m2 |
Aeff | Normalized effective projected area | m2 |
AI | Arching index | — |
Ap | Cross-sectional area of plate | m2 |
Aproj | Projected area of indent | m2 |
AR | Aspect ratio | — |
ARST | Cross-sectional area of Schulze ring shear cell | m2 |
ATD | Average time difference | s |
AU | Cross-sectional area of uniaxial compaction tester | m2 |
AJT | Cross-sectional area of Jenike cell | m2 |
b | Body force | N |
B | Distance of barycentre of mass from axis of rotation | m |
BIM | Ball indentation method | — |
Bo | Bond number | — |
c | Cohesion | Pa |
C | Constraint factor | — |
CBeverloo | Flowability index | — |
CI | Carr index | — |
Co | Compressibility | — |
Coh | Cohesion number | — |
d | Particle size | m |
d43 | Volume moment mean diameter | m |
db | Indenter diameter | m |
dc | Diameter of contact | m |
dFe | Feret diameter | m |
dFe⊥ | Feret diameter measured in perpendicular direction | m |
dv | Particle equivalent volume sphere diameter | m |
D | Aperture diameter | m |
Dheap | Diameter of powder heap | m |
e | Rod diameter | m |
eeff | Effective coefficient of restitution | — |
E | Young modulus | Pa |
Ef | Flow energy | J |
f | Inter-particle force | N |
fc | Unconfined yield stress | Pa |
ff | Flow-factor | — |
F | Force | N |
Fc | Contact force | N |
FD | Fractal dimensions | — |
Fg | Gravitational force | N |
F | Contact force exerted on i by j | N |
Fn | Normal load | N |
Fnc | Non-contact force | N |
Fr | Frounde number | — |
Fs | Macroscopic shear force | N |
Ft | Translational force | N |
g | Gravitational acceleration | m s−2 |
g0 | Radial distribution function | — |
Gn | Solid fraction dependent functions in rapid flow regime | — |
h | Height | m |
hB | Burke constant | — |
hc,max | Indentation unloading intercept with depth axis | m |
hd | Indentation depth | m |
hf | Final indentation depth | m |
hsep | Separation distance | m |
H | Indentation hardness | Pa |
HR | Haussner ratio | — |
i | Element under consideration | — |
I | Moment of inertia/inertial number | kg m2 |
I0 | Constant for μ(I) rheology | — |
IA, IB | Modified Bessel functions of the first kind | — |
k | Permeability | m2 |
kA | Fraction of collisional dissipation contribution to shear viscosity | — |
kB | Fraction of frictional contribution to shear viscosity | — |
kc | Carman constant | — |
kj | Constant for Johanson hang-up indiciser | ft |
Ka | Active Rankine coefficient | — |
Kp | Passive Rankine coefficient | — |
l | Rod length | m |
lij | Vector from mass centre of particle i to mass centre of particle j | m |
ℓp | Unit vector along the major axis | — |
L | Drum length | m |
Lp | Perimeter length | m |
Ly | Dimension of domain in y-direction | m |
m | Mass | kg |
M | Contact torque | N m |
n | Number of particles | — |
N | Macroscopic normal force | N |
Nc | Applied force at failure | N |
O | Order parameter | — |
p | Particle under consideration | — |
P | Pressure | Pa |
Pa | Active pressure | Pa |
Pp | Passive pressure | Pa |
PFI | Powder flow index | — |
PR | Packing ratio | — |
q | Energy flux | N m−1 s−1 |
Qij | Symmetric traceless order tensor | Pa |
r | Asperity size | m |
rcol | Radius of meniscus curvature | m |
rro | Surface roughness length | m |
R | Radius | m |
R1 | Minor radius of the lid cross-sectional annular area | m |
R2 | Major radius of the lid cross-sectional annular area | m |
Rgas | Ideal gas constant | kg m s−2 K−1 |
RH | Relative humidity | % |
RI | Ratholing index | — |
s | Particle surface area | m2 |
S | Stiffness | N m−1 |
Seff | Effective projected surface area of a particle | m2 |
Ssph | Effective projected surface area of the sphere of the same volume to the particle | m2 |
STD | Standard deviation | — |
t | Time | s |
T | Torque | N m |
Tabs | Temperature | K |
Tgran | Granular temperature | m2 s−2 |
U | Velocity at boundary | m s−1 |
Ucons | Consolidation velocity | m s−1 |
Umf | Minimum fluidisation velocity | m s−1 |
Uro | Rain-off velocity | m s−1 |
Ut | Terminal velocity | m s−1 |
v | Velocity | m s−1 |
vax | Axial velocity | m s−1 |
〈vp,ave〉 | Average particle velocity | m s−1 |
vp,fl | Fluctuating velocity | m s−1 |
vp,i | Velocity of particle i | m s−1 |
V | Particle volume | m3 |
Vliq | Volume of liquid per contact | m3 |
Vm | Molar volume | m3 |
w | Weight | N |
W | Mass flow rate | kg s−1 |
x | Distance between crossbeam holders | m |
y | Parameter of yield locus curvature | — |
Y | Yield stress | Pa |
Z | Height | m |
Greek Characters
α | Angle of inclination | rad |
〈α〉 | Particle angularity | — |
αd | Dynamic angle of repose | rad |
αS | Static angle of repose | rad |
β | Angle between the powder surface and horizontal line | rad |
βn | Fitting parameter | — |
γ | Surface tension or energy | J m−2 |
γ0 | Fitting parameter | — |
γ°* | Dimensionless shear rate | — |
| Shear strain rate | s−1 |
Γ | Energy dissipated per unit volume and unit time | Pa s−1 |
δ | Filling angle | rad |
δij | Kronecker delta in the continuum mechanics notation | — |
ΔP | Pressure drop | Pa |
Δγ | Work of adhesion | J |
ϵ | Internal energy per unit volume | J m−3 |
ɛ | Porosity | — |
ɛ0 | Permittivity of free space | — |
ɛr | Dielectric constant of material | — |
ζ | Dissipation function | — |
η | Dynamic viscosity | Pa s |
ηs | Shear viscosity | Pa s |
ηs,fr | Frictional shear viscosity | Pa s |
θ | Angle of inclination | rad |
θwet | Wetting angle | rad |
κ | Coordination number | — |
κc | Surface curvature | m−1 |
λ | Cohesive length | m |
λσ | Horizontal to vertical stress ratio | — |
μ | Friction coefficient | — |
μd | Dynamic friction coefficient | — |
μ(I) | Friction coefficient under given inertial number | — |
μk | Kinematic friction coefficient | — |
μr | Rolling friction coefficient | — |
μs | Static friction coefficient | — |
μ2 | Upper limit of the friction coefficient | — |
ν | Poisson ratio | — |
ξ | Standard deviation of velocity fluctuations | m2 s−1 |
ρ | Density | kg m−3 |
ρa | Air density | kg m−3 |
ρaer | Aerated density | kg m−3 |
ρb | Bulk density | kg m−3 |
ρf | Fluid density | kg m−3 |
ρp | Particle density | kg m−3 |
ρpour | Poured density | kg m−3 |
ρs | Solid (true) density | kg m−3 |
ρtap | Tapped density | kg m−3 |
σ | Cauchy stress tensor | Pa |
σ | Normal stress | Pa |
σ0 | Fitting parameter | — |
σ1 | Major principal stress | Pa |
σ1,b | Maximum consolidation stress at bed bottom | Pa |
σ2 | Minor principal stress (in 2D) | Pa |
σf | Dynamic cohesion index | — |
σkin | Kinematic stress | Pa |
σm | Normal stress | Pa |
σm0 | Normal stress at top of powder bed increment | Pa |
σp | Pre-shear normal stress | Pa |
σs | Applied normal stress during shearing | Pa |
σt | Tensile strength | Pa |
| Centre of consolidation Mohr circle on abscissa | Pa |
Stress matrix in plane deformation | Pa | |
τ | Shear stress | Pa |
τp | Steady state shear stress | Pa |
τs | Corrected shear stress at incipient flow | Pa |
τs′ | Shear stress at incipient flow | Pa |
ϕ | Angle of inclination | rad |
ϕe | Effective angle of internal friction | rad |
ϕi | Static angle of internal friction | rad |
ϕw | Angle of wall friction | rad |
ϕw′ | Slope of angle of wall yield locus | rad |
Φ | Angle of repose | rad |
χ | Solid fraction | — |
ψ | Sphericity factor | — |
ω | Rotational velocity | rad s−1 |
α | Angle of inclination | rad |
〈α〉 | Particle angularity | — |
αd | Dynamic angle of repose | rad |
αS | Static angle of repose | rad |
β | Angle between the powder surface and horizontal line | rad |
βn | Fitting parameter | — |
γ | Surface tension or energy | J m−2 |
γ0 | Fitting parameter | — |
γ°* | Dimensionless shear rate | — |
| Shear strain rate | s−1 |
Γ | Energy dissipated per unit volume and unit time | Pa s−1 |
δ | Filling angle | rad |
δij | Kronecker delta in the continuum mechanics notation | — |
ΔP | Pressure drop | Pa |
Δγ | Work of adhesion | J |
ϵ | Internal energy per unit volume | J m−3 |
ɛ | Porosity | — |
ɛ0 | Permittivity of free space | — |
ɛr | Dielectric constant of material | — |
ζ | Dissipation function | — |
η | Dynamic viscosity | Pa s |
ηs | Shear viscosity | Pa s |
ηs,fr | Frictional shear viscosity | Pa s |
θ | Angle of inclination | rad |
θwet | Wetting angle | rad |
κ | Coordination number | — |
κc | Surface curvature | m−1 |
λ | Cohesive length | m |
λσ | Horizontal to vertical stress ratio | — |
μ | Friction coefficient | — |
μd | Dynamic friction coefficient | — |
μ(I) | Friction coefficient under given inertial number | — |
μk | Kinematic friction coefficient | — |
μr | Rolling friction coefficient | — |
μs | Static friction coefficient | — |
μ2 | Upper limit of the friction coefficient | — |
ν | Poisson ratio | — |
ξ | Standard deviation of velocity fluctuations | m2 s−1 |
ρ | Density | kg m−3 |
ρa | Air density | kg m−3 |
ρaer | Aerated density | kg m−3 |
ρb | Bulk density | kg m−3 |
ρf | Fluid density | kg m−3 |
ρp | Particle density | kg m−3 |
ρpour | Poured density | kg m−3 |
ρs | Solid (true) density | kg m−3 |
ρtap | Tapped density | kg m−3 |
σ | Cauchy stress tensor | Pa |
σ | Normal stress | Pa |
σ0 | Fitting parameter | — |
σ1 | Major principal stress | Pa |
σ1,b | Maximum consolidation stress at bed bottom | Pa |
σ2 | Minor principal stress (in 2D) | Pa |
σf | Dynamic cohesion index | — |
σkin | Kinematic stress | Pa |
σm | Normal stress | Pa |
σm0 | Normal stress at top of powder bed increment | Pa |
σp | Pre-shear normal stress | Pa |
σs | Applied normal stress during shearing | Pa |
σt | Tensile strength | Pa |
| Centre of consolidation Mohr circle on abscissa | Pa |
Stress matrix in plane deformation | Pa | |
τ | Shear stress | Pa |
τp | Steady state shear stress | Pa |
τs | Corrected shear stress at incipient flow | Pa |
τs′ | Shear stress at incipient flow | Pa |
ϕ | Angle of inclination | rad |
ϕe | Effective angle of internal friction | rad |
ϕi | Static angle of internal friction | rad |
ϕw | Angle of wall friction | rad |
ϕw′ | Slope of angle of wall yield locus | rad |
Φ | Angle of repose | rad |
χ | Solid fraction | — |
ψ | Sphericity factor | — |
ω | Rotational velocity | rad s−1 |
Subscripts
10 | 10th percentile | — |
50 | 50th percentile | — |
90 | 90th percentile | — |
ave. | Average | — |
b | Bed | — |
col | Collisional | — |
d | Drag force or of equivalent drag force | — |
eff | Effective | — |
end | At the end of the test | — |
eq | Equivalent | — |
fl | Fluctuating | — |
i | Case under consideration | — |
kin | Kinetic | — |
max | Maximum | — |
min | Minimum | — |
plug | Of plug | — |
s | Of equivalent surface area | — |
sat | Saturation | — |
sv | Of equivalent specific surface area | — |
tv | Of equivalent terminal velocity | — |
v | Of equivalent volume | — |
w | Wall | — |
x | x component | — |
y | y component | — |
Ω | Of equivalent projected surface area | — |
10 | 10th percentile | — |
50 | 50th percentile | — |
90 | 90th percentile | — |
ave. | Average | — |
b | Bed | — |
col | Collisional | — |
d | Drag force or of equivalent drag force | — |
eff | Effective | — |
end | At the end of the test | — |
eq | Equivalent | — |
fl | Fluctuating | — |
i | Case under consideration | — |
kin | Kinetic | — |
max | Maximum | — |
min | Minimum | — |
plug | Of plug | — |
s | Of equivalent surface area | — |
sat | Saturation | — |
sv | Of equivalent specific surface area | — |
tv | Of equivalent terminal velocity | — |
v | Of equivalent volume | — |
w | Wall | — |
x | x component | — |
y | y component | — |
Ω | Of equivalent projected surface area | — |