Symbols in Bulk Solids Technology

Overview of symbols for calculations and measurements. Deviations from the notation according to DIN EN 1991-4 are marked accordingly.

Symbol	DIN EN 1991-4	Unit	Description
a	a	m	Distance from transition to outlet (hopper)
b	b	m	Width of outlet opening (rectangular cross-section)
d_c	d_c	m	Diameter of circular outlet opening
h	h	m	Height of vertical section
h_b	h_b	m	Height of hopper
h_c	h_c	m	Equivalent surface height
h_h	h_h	m	Hopper height
h_o	h_o	m	Characteristic height, reference height
z	z	m	Vertical coordinate
α	α	°	Hopper half angle
β	β	°	Wall slope angle
A	A	m²	Cross-sectional area
U	U	m	Perimeter of cross-section
K	K	-	Horizontal load ratio (lateral pressure ratio)
P_h	p_h	kN/m²	Horizontal pressure
P_v	p_v	kN/m²	Vertical pressure
φ_w	φ_w	°	Wall friction angle
μ	μ	-	Wall friction coefficient
b_min	b_min	m	Minimum outlet width
DRH	—	m	Hydraulic diameter ratio

Symbol	DIN EN 1991-4	Unit	Description
σ_r	—	Pa	Reference Stress
σ_N	σ_N	Pa	Normal Stress
σ₁	σ₁	Pa	Major Principal Stress
σ₂	σ₂	Pa	Minor Principal Stress (also σ₃)
σ_c	σ_c	Pa	Unconfined Compressive Strength
σ_t	—	Pa	Tensile Strength
τ_c	c	Pa	Cohesion (≙ c at DIN EN 1991-4)
τ_rm	—	Pa	Pre-Shear Stress
τ_m	—	Pa	Shear Stress
ρ_b0	—	kg/m³	Loose Bulk Density
ρ_br	—	kg/m³	Bulk Density
FL	—	-	Flowability Factor = σ₁ / σ_c
FLR	—	-	Relative Flowability Factor = ( σ₁ - σ₂ ) / σ_c
FLA	—	-	Absolute Flowability Factor = FLR * ρ_br
φ_i	Φ_c	°	Angle of Internal Friction
φ_e	Φ_i	°	Effective Angle of Friction (pre-shear) (≙ Φ_i at EN1991-4) ⇒ K
φ_s	—	°	Angle of friction at steady-state flow
K₀	λ₀	-	Coefficient of earth pressure at rest (K₀) = σ₂ / σ₁ = (1 - sin φ_e) / (1 + sin φ_e)
K_m	K	-	Mean Lateral Pressure Ratio = 1.1 * (1 - sin φ<sub>e</sub>) according to EN 1991-4
R	—	-	Correlation Coefficient of Yield Locus
τ_s	—	Pa	Dynamic Shear Stress at normal stress
τ_s,c	—	Pa	Cohesion of Friction at steady-state flow
σ_s,1	—	Pa	Major Principal Stress at steady-state flow
σ_s,2	—	Pa	Minor Principal Stress at steady-state flow
R_s	—	-	Correlation Coefficient of Friction at steady-state flow

Symbol	DIN EN 1991-4	Unit	Description
σ_w	σ_w	Pa	Normal stress at the wall
τ_w,s	—	Pa	Shear stress at the wall (static)
τ_w,d	—	Pa	Shear stress at the wall (dynamic)
μ_s	—	-	Wall friction coefficient (static)
μ_d	—	-	Wall friction coefficient (dynamic)
φ_s	—	°	Wall friction angle (static)
φ_d	—	°	Wall friction angle (dynamic)
τ_s	—	Pa	Shear strength (static)
τ_d	—	Pa	Shear strength (dynamic)

Symbol	DIN EN 1991-4	Unit	Description
ρ_b	ρ_b	kg/m³	Bulk density
ρ_b0	—	kg/m³	Bulk density at rest (after consolidation)
ρ(σ)	—	kg/m³	Bulk density as function of consolidation stress
ρ(σ)_Log	—	kg/m³	Bulk density (logarithmic function)
ρ(σ)_Pow	—	kg/m³	Bulk density (power function)
R_Log	—	-	Correlation coefficient (logarithmic)
R_Pow	—	-	Correlation coefficient (power)
CI_Log	—	%	Compressibility index (logarithmic)
HR_Log	—	%	Hausner ratio (logarithmic)

Symbol	DIN EN 1991-4	Unit	Description
Φ_AR	—	°	Angle of repose
h_peak	—	mm	Height of peak (cone height)
d_outside	—	mm	Outside diameter of cone base
Φ_AR,avg	—	°	Average angle of repose
SD_AR	—	°	Standard deviation of angle of repose

DIN EN 1991-4 - Eurocode 1: Actions on structures - Part 4: Silos and tanks
ASTM D6682 - Standard Test Method for Determining the Flow Properties of Powders
DIN ISO 4324 - Surface active agents - Powders and granules - Determination of the angle of repose