Symbols
Symbols in bulk solids technology
Overview of the symbols for calculations and measurements. Where EN 1991-4 uses a different symbol, it is given in the second column.
Silo calculation
Symbols for the design of silos to EN 1991-4
| Symbol | EN 1991-4 | Unit | Description |
|---|---|---|---|
| a | - | m | Side length of a silo with rectangular or hexagonal cross-section |
| b | - | m | Width of a rectangular silo |
| dc | - | m | Characteristic dimension of the cross-section of the vertical silo section: dc = 4A/U |
| h | - | m | Vertical depth below the transition |
| hb | - | m | Total height of a silo from the virtual hopper apex to the equivalent surface |
| hc | - | m | Height of the vertical silo section from the transition to the equivalent surface |
| hh | - | m | Hopper height from the virtual hopper apex to the transition |
| ho | - | m | Vertical distance between the equivalent surface and the lowest point of the solid surface (base of the surface cone) |
| z | - | m | Depth below the equivalent surface of the solid in the filled condition |
| α | - | ° | Inclination of the hopper wall to the horizontal (α = 90° − β) |
| β | - | ° | Inclination of the hopper wall to the vertical; for rectangular hoppers β is the steepest wall angle |
| A | - | m² | Horizontal cross-sectional area of the vertical silo section |
| U | - | m | Internal perimeter of the cross-section of the vertical silo section |
| K | - | - | Characteristic lateral pressure ratio: K = ph / pv (dimensionless) |
| ph | - | kN/m² | Horizontal pressure of the stored solid at rest on the silo wall |
| pv | - | kN/m² | Vertical pressure in the stored solid |
| φw | - | ° | Wall friction angle on the vertical silo wall |
| μ | - | - | Wall friction coefficient: μ = tan(φw) (dimensionless) |
| bmin | - | m | Minimum outlet opening to prevent arching |
| DRH | - | m | Rathole diameter to prevent ratholing |
Yield locus
Symbols for yield locus and flowability, measured to ASTM D6682-A
| Symbol | EN 1991-4 | Unit | Description |
|---|---|---|---|
| σr | — | Pa | Reference stress (pre-consolidation stress) |
| σN | σN | Pa | Normal stress |
| σ1 | σ1 | Pa | Major principal stress |
| σ2 | σ2 | Pa | Minor principal stress |
| σc | σc | Pa | Unconfined compressive strength |
| σt | — | Pa | Tensile strength |
| σs,1 | — | Pa | Major principal stress at steady-state flow |
| σs,2 | — | Pa | Minor principal stress at steady-state flow |
| τc | c | Pa | Cohesion (≙ c in EN 1991-4) |
| τrm | — | Pa | Pre-shear stress |
| τm | — | Pa | Shear stress at incipient flow |
| τs | — | Pa | Shear stress at steady-state flow at normal stress |
| τs,c | — | Pa | Cohesion at steady-state flow |
| ρb0 | — | kg/m³ | Bulk density (loose) |
| ρbr | — | kg/m³ | Bulk density (consolidated) |
| FL | — | - | Flowability = σ1 / σc |
| FLR | — | - | Relative flowability = (σ1 − σ2) / σc |
| FLA | — | - | Absolute flowability = FLR · ρbr, relative to the density at consolidation stress |
| φi | φc | ° | Angle of internal friction (≙ φc in EN 1991-4) |
| φe | φi | ° | Effective angle of internal friction (pre-shear) (≙ φi in EN 1991-4) |
| φs | — | ° | Angle of internal friction at steady-state flow |
| K0 | λ0 | - | Coefficient of earth pressure at rest = σ2 / σ1 |
| Km | K | - | Mean lateral pressure ratio = 1.1 · (1 − sin φe) to EN 1991-4 |
| R | — | - | Correlation coefficient of the yield locus |
| Rs | — | - | Correlation coefficient of steady-state flow |
Wall friction
Symbols for wall friction, measured to ASTM D6682-B
| Symbol | EN 1991-4 | Unit | Description |
|---|---|---|---|
| σw | - | Pa | Normal stress perpendicular to the wall surface in the wall friction test |
| τw,s | - | Pa | Wall shear stress at static friction (start-up from rest) |
| τw,d | - | Pa | Wall shear stress at dynamic friction |
| μs | - | - | Wall friction coefficient at static friction: μs = tan(φs) (dimensionless) |
| μd | μ | - | Wall friction coefficient at dynamic friction: μd = tan(φd) (dimensionless) |
| φs | - | ° | Wall friction angle at static friction |
| φd | φw | ° | Wall friction angle at dynamic friction |
| φx | - | ° | Local wall friction angle: tan(φx) = τw / σw, slope of the straight line from the origin to the wall yield locus at the respective normal stress |
| τs | - | Pa | Adhesion stress at static friction |
| τd | - | Pa | Adhesion stress at dynamic friction |
Bulk density
Symbols for bulk and compression density, measured to EN ISO 60 or ASTM D6682-C
| Symbol | EN 1991-4 | Unit | Description |
|---|---|---|---|
| ρb | - | kg/m³ | Bulk density |
| ρb0 | - | kg/m³ | Loose bulk density without mechanical compaction |
| ρ(σ) | - | kg/m³ | Bulk density as a function of the consolidation stress |
| ρ(σ)Log | - | kg/m³ | Logarithmic density function: ρ(σ) = b·ln(σ) + a |
| ρ(σ)Pow | - | kg/m³ | Power function of the compression density: ρ(σ) = a·σb |
| RLog | - | - | Correlation coefficient R of the logarithmic fit (dimensionless) |
| RPow | - | - | Correlation coefficient R of the power fit (dimensionless) |
| CILog | - | % | Carr compressibility index from the logarithmic function |
| HRLog | - | - | Hausner ratio from the logarithmic function (dimensionless) |
Angle of repose
Symbols for the angle of repose, measured to DIN ISO 4324 or ASTM D6393
| Symbol | EN 1991-4 | Unit | Description |
|---|---|---|---|
| φAR | φr | ° | Angle of repose of a conical heap to DIN ISO 4324, determined as φAR = arctan(2hpeak / doutside) |
| hpeak | - | mm | Vertical height of the cone tip above the base |
| doutside | - | mm | Diameter of the circular base of the cone |
| φAR,avg | - | ° | Arithmetic mean of n individual measurements of the angle of repose |
| SDAR | - | ° | Empirical standard deviation of the angle of repose test series |
Moisture content
Symbols for moisture determination by the oven-drying method or with a halogen moisture analyser
| Symbol | EN 1991-4 | Unit | Description |
|---|---|---|---|
| m0 | — | g | Initial mass of the moist sample |
| mdry | — | g | Dry mass after drying to constant mass |
| %M | — | % | Moisture relative to the moist sample: %M = 100 · (m0 − mdry) / m0 |
| %S | — | % | Solids content relative to the moist sample: %S = 100 · mdry / m0 |
| %Mdb | — | % | Moisture on a dry basis: 100 · (m0 − mdry) / mdry |
| %Sdb | — | % | Solids on a dry basis: 100 · m0 / mdry |
Time consolidation
Symbols for the time consolidation of yield locus and wall friction
| Symbol | EN 1991-4 | Unit | Description |
|---|---|---|---|
| t | — | h | Storage time under load (time consolidation) |
| f | — | - | Time consolidation factor: ratio of the shear stress after the storage time to the shear stress without storage time (dimensionless, ≥ 1) |
| Δτc | — | Pa | Increase in cohesion due to time consolidation: Δτc = (f − 1) · τ, where τ is the shear stress at the pre-shear point |
| σc,t | — | Pa | Unconfined compressive strength after storage time |
| σc,t(σ1) | — | Pa | Time flow function: unconfined compressive strength after storage time as a function of the consolidation stress |
| ffc,t | — | - | Flowability after storage time: ffc,t = σ1 / σc,t |
| Δτc,w | — | Pa | Increase in wall adhesion due to time consolidation on the wall |
Referenced standards
- EN 1991-4 – Eurocode 1: Actions on structures – Part 4: Silos and tanks
- ASTM D6682 – Measuring Shear Stresses of Powders Using Rotational Split Level Shear Tester (A: yield locus, B: wall friction, C: compression density)
- EN ISO 60 – Plastics – Determination of apparent density of material that can be poured from a specified funnel (bulk density)
- USP 616 – Bulk Density and Tapped Density of Powders, Method III (bulk density)
- DIN ISO 4324 – Surface active agents – Powders and granules – Measurement of the angle of repose
- ASTM D6393 – Standard Test Method for Bulk Solids Characterization by Carr Indices (angle of repose)
- USP 1174 – Powder Flow (classification by angle of repose, Carr index and Hausner ratio)