Density of bulk solids: measurement and parameters
Bulk solids are compressible: under load the particles rearrange, the voids decrease and the density rises. In the silo the density therefore increases with depth, and a bulk solid once compacted largely retains its packing after unloading.
From bulk density, compression density and flow density you obtain the density function ρb(σ) – the basis for the bulk unit weight γ to EN 1991-4, the filling volume and the evaluation of the flow function.
Parameters from the density measurement
The density is measured from the loose state up to the stresses in the silo.
| Parameter | Symbol | in EN 1991-4 | Use |
|---|---|---|---|
| Bulk density | ρb0 | – | filling volume, storage and transport |
| Compression density | ρb(σ) | γ = ρb · g | silo loads (bulk unit weight) |
| Flow density | ρb(σ1) | – | evaluation of the yield loci |
| Density function | ρb(σ) | – | density at every stress in the silo, arching |
| Hausner ratio | H = ρb / ρb0 | – | comparing compressibility |
| Carr index | C = (1 − ρb0 / ρb) · 100 % | – | comparing compressibility |
Carrying out the density measurement
The bulk density is determined to EN ISO 60 or USP 616 III, the compression density on standardised shear testers to ASTM D6682-C and the flow density during the yield locus measurement to ASTM D6682-A.
- Bulk density The sample is poured loosely into a measuring vessel under defined conditions and weighed.
- Compression In the measuring cell the sample is loaded vertically in steps – uniaxial compaction – and the density is determined for each step.
- Flow density During the yield locus measurement the density after consolidation is measured as well – multiaxial compaction.
- Density function A fitted function over the measuring points gives the density as a function of the stress.
Stress steps
The stress steps cover the range from the loose bulk solid up to the largest pressure in the silo. At each step the load is held until the density no longer changes. Each measurement is carried out at least in duplicate.
Density function
The measuring points are fitted by a logarithmic regression ρb(σ) = b · ln σ + a or a power regression ρb(σ) = a · σb. This makes the density available for every stress – also between the measuring points, for example for the bulk unit weight at the maximum filling pressure or for the weight of a bulk solid arch at the outlet. As pressure and density depend on each other, the density at the respective location in the silo is usually determined iteratively.
Test report and customer portal
The test report contains the pure measured values and both fitted functions with correlation coefficient and standard deviation, as well as Carr index, Hausner ratio and conversion factor. In the customer portal, different measurements can be set side by side and compared.
Uniaxial and multiaxial compaction
Compressibility
Unlike a liquid, which is approximately incompressible, a bulk solid compacts under load: the particles rearrange, the porosity decreases and more contact points form between the particles. The adhesive forces grow with them – with stronger consolidation, the shear stress required for flow therefore rises.
Compaction is largely irreversible: after the load is removed, the packing is largely retained. A bulk solid that has once been under pressure is denser than in the freshly poured state.
Bulk unit weight to EN 1991-4
EN 1991-4 uses the compression density, i.e. the density under uniaxial compaction. The bulk unit weight γ is determined at the stress at the bottom of the vertical silo section – where the vertical filling pressure, and thus the density, is greatest. The characteristic value is given by the conversion factor aγ. For a more precise analysis the density is determined from the density function at each location to be calculated.
Flow density
The flow density is the density from the shear test after consolidation. Multiaxial compaction rearranges the particles further. It is determined with every yield locus measurement. Even though EN 1991-4 only considers uniaxial compaction, both densities can be used in the evaluation.
Hausner ratio and Carr index
Both indices describe how strongly a bulk solid can be compacted. They are calculated from the bulk density and the fitted density function and are suitable for comparing bulk solids, batches or formulations. The usual classification to USP 1174:
| Carr index C | Hausner ratio H | Flow character |
|---|---|---|
| ≤ 10 % | 1.00 – 1.11 | excellent |
| 11 – 15 % | 1.12 – 1.18 | good |
| 16 – 20 % | 1.19 – 1.25 | fair |
| 21 – 25 % | 1.26 – 1.34 | passable |
| 26 – 31 % | 1.35 – 1.45 | poor |
| 32 – 37 % | 1.46 – 1.59 | very poor |
| ≥ 38 % | ≥ 1.60 | very, very poor |
For silo design and the assessment of flow problems, the Hausner ratio and the Carr index do not replace the measurement of the yield locus and the flowability ffc.
Frequently asked questions about density measurement
- Which density is used for the silo loads?
- EN 1991-4 uses the bulk unit weight γ from the compression density, i.e. the density under uniaxial compaction. The stress at the bottom of the vertical silo section, where the density is greatest, is decisive. For a more precise analysis the density is determined from the density function at each location to be calculated – usually iteratively, as pressure and density depend on each other.
- What is the difference between bulk, compression and flow density?
- The bulk density ρb0 applies to the loosely poured, uncompacted bulk solid. The compression density is the density under uniaxial compaction; it is also used in EN 1991-4. The flow density is the density from the shear test after consolidation, i.e. under multiaxial compaction. Here the particles additionally rearrange; the flow density is therefore usually somewhat higher than the compression density at the same stress.
- Are the Hausner ratio and the Carr index also determined?
- Yes. Both indices are calculated from the bulk density and the fitted density function and are stated in the test report together with the classification.
- How much sample material is needed?
- About 200 ml per sample preparation are needed for the density measurement; each measurement is carried out at least in duplicate. If yield loci or wall friction are measured as well, the quantity needed for them is added.
- How long does the measurement take?
- Results are usually available about two weeks after sample receipt – depending on laboratory capacity and the number of samples. Time consolidation tests can extend the standard lead time accordingly.
Have the density measured
Describe your bulk solid and the application – we define the stress steps and prepare a non-binding quote for you.
- Bulk density, compression density and density function ρb(σ)
- Hausner ratio and Carr index
- Results online in the customer portal and as a test report