Flowability of bulk solids: flowability factor and flow function
The flowability factor ffc = σ1 / σc relates the consolidation stress to the unconfined compressive strength of the consolidated bulk solid. It is the most widely used parameter for describing the flowability of powders and bulk solids in a single value.
For quality control, product release and product optimisation: compare batches, deliveries and formulations at the same consolidation stress – and dimension the outlet and rathole diameter in the silo with the flow function.
Flowability parameters
All parameters come from the yield locus measurement – together with internal friction, cohesion and unconfined compressive strength.
| Parameter | Symbol | Use |
|---|---|---|
| Flowability factor | ffc = FL = σ1 / σc | classification of flowability, comparison at the same σ1 |
| Relative flowability factor | FLR = (σ1 − σ2) / σc | supplement for pronounced internal friction |
| Absolute flowability factor | FLA = FLR · ρbr | comparing bulk solids of different density |
| Flow function | σc(σ1) | arching, ratholing |
| Time flow function | σc,t(σ1) | flowability after storage time |
Classification by the flowability factor
The larger ffc, the better the bulk solid flows. The classification corresponds to that in the test report and the online evaluation:
Comparison at the same consolidation
As the unconfined compressive strength σc always depends on the state of consolidation, comparative values are only meaningful if they were determined at the same consolidation stress σ1. Bulk solids with a similar ffc then show similar flow behaviour in the application.
Classification in context
The flowability factor allows a quick classification but responds non-linearly to the underlying measured values. It is therefore assessed together with the other parameters of the yield locus and the application – for low values with a check of dischargeability or processability.
Types of bulk solid by the yield locus
Behind the classes of the classification lies the shape of the yield locus: by cohesion τc and angle of internal friction φi, bulk solids can be assigned to four limiting cases – with different consequences for storage, discharge and transport. Plastic, cohesive and cohesionless bulk solids appear on the scale; fluidised bulk solids have no unconfined compressive strength and lie outside the scale.
1 · Plastic bulk solids (φi = 0, τc > 0)
Ideally plastic bulk solids are incompressible. Owing to saturation, their particles cannot change position; the contact areas remain the same despite changing normal stress. The largest transmissible shear stress therefore depends only on the cohesion, τ = τc (e.g. saturated clay). Some samples only show plastic behaviour at higher normal stresses.
Owing to their usually high cohesion, such bulk solids are difficult to handle.
2 · Cohesive bulk solids (φi > 0, τc > 0)
Because of adhesive forces at the particle contacts, they resist shear stresses even at a normal stress σ = 0; friction and cohesion act together: τ = σ · tan φi + τc. The rearrangement of the particles at rest is restricted by the cohesion – previous compaction by normal and shear stress as well as time consolidation also determine the later flow behaviour. The angle of repose therefore also depends on the history. Measurements at or near the bulk density ρb0 allow no conclusions about the storage and flow behaviour under load; measurements in the shear tester are meaningful.
During flow, the distance between the particles increases and the cohesive forces largely lose their influence. The particles can rearrange again, the packing density in the shear zone changes, and time consolidation can also be released again.
This group is the most common in practice. The unconfined compressive strength can lead to arching and thus to discharge problems if hopper inclination and outlet opening are not adequately dimensioned; possible time consolidation has to be taken into account, as does the scatter of the material properties.
3 · Cohesionless bulk solids (φi > 0, τc = 0)
Cohesionless, also free-flowing, bulk solids follow the straight line τ = σ · tan φi (e.g. dry sand). Their yield locus passes through the origin, and the angle of internal friction corresponds to the angle of repose. The frictional forces depend only on the applied normal stress, not on the history.
These bulk solids are usually easy to handle: without unconfined compressive strength no arches form at the outlet, and preloading is reversible. For the design of the storage, the bulk density under load and the lateral pressure ratio are determined.
4 · Fluidised bulk solids (φi = 0, τc = 0)
Owing to large particle distances, fluidised bulk solids are cohesionless and without particle friction – they behave like a liquid. The gas usually escapes again more or less quickly, and a non-fluidised state sets in. Fine, light materials in particular tend towards fluidisation.
During rehandling and transport, for example through mixing with air, such states can occur and let the bulk solid flush through shut-off and conveying devices. Sufficient deaeration and dwell time before further processing prevents this; special precautions are to be taken when plants are filled for the first time.
Relative and absolute flowability factor
Relative flowability factor FLR
The classic flowability factor is not equally meaningful for all bulk solids. The relative flowability factor FLR = (σ1 − σ2) / σc relates the difference between the major and minor principal stress to the unconfined compressive strength. It responds particularly to bulk solids with pronounced internal friction and supplements ffc.
Absolute flowability factor FLA
To compare bulk solids of different density better, the absolute flowability factor multiplies the relative one by the density at consolidation stress: FLA = FLR · ρbr. In practice, FLA is often used instead of FLR.
Comparing batches and products
For product changes or changing batches of raw materials and products, the comparison shows whether a bulk solid can still be processed in the existing plant. In the customer portal, different measurements can be set side by side and compared.
Flow function over the stress range
To represent flowability over a larger stress range, yield loci are measured at a minimum of three different consolidation stresses covering the range of application. The unconfined compressive strength as a function of the consolidation stress gives the flow function σc(σ1). It is the basis for technological calculations such as the critical outlet diameter against arching and the rathole diameter against ratholing.
Course of the flow function
The flow function is often represented by a straight line, which simplifies further calculations. At low stresses, however, it frequently shows a curved course – also because the unconfined compressive strength σc cannot be greater than the major principal stress σ1.
At higher stresses, saturation effects, degradation or attrition, but also agglomeration, can change the flow properties. Measurements over a larger stress range make such effects visible.
Methods for determining flowability
Several methods have become established in practice. They differ above all in whether they describe the loose or the consolidated bulk solid.
| Method | State of the bulk solid | Suitability | Effort |
|---|---|---|---|
| Flow through an orifice (flow time through a standardised opening) | loosely poured | cohesionless to slightly cohesive bulk solids | low |
| Angle of repose | loosely poured | indicator for loose pouring; poorly reproducible at high cohesion | low |
| Yield locus and ffc | consolidated, depending on the stress level | all bulk solids; also provides internal friction, cohesion, unconfined compressive strength and time consolidation | higher |
Frequently asked questions about flowability
- How much sample material is needed for flowability?
- About 0.5 litres is sufficient for a flowability measurement alone at one consolidation stress; for the flow function for silo design about 1 to 2 litres per sample are needed.
- At which consolidation stress is flowability compared?
- The unconfined compressive strength σc always depends on the consolidation. Comparative values are therefore only meaningful if they were determined at the same consolidation stress σ1. We define the stress on the basis of the application – for batch and product comparisons it remains the same for all samples.
- Is the flowability factor sufficient for silo design?
- No. ffc classifies the flowability at one stress. For outlet, rathole and hopper inclination, the flow function over the stress range of the silo, the effective angle of internal friction and the wall friction are needed.
- Which bulk solids are critical?
- Above all cohesive bulk solids: their unconfined compressive strength can lead to arches and ratholes at the outlet if hopper inclination and outlet opening are not suitably dimensioned; possible time consolidation adds to this. Fluidised bulk solids can flush through shut-off and conveying devices.
- 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 flowability measured
Describe your bulk solid and the comparison you need – we define the consolidation stress and prepare a non-binding quote for you.
- ffc, FLR and FLA at a defined consolidation stress
- Flow function for silo design, time consolidation on request
- Results online in the customer portal and as a test report