A Review Of Qualitative Versus Quantitative Bulk Flow Property Tests

Introduction

This article provides a review of the wide range of different bulk solid flow and characterisation tests and categorises them as either qualitative tests that maybe useful if test represents your process, or quantitative tests that measure fundamental bulk properties.  

Generally, the flow properties of bulk solids are strongly dependent on the prior stress history.  Thus, a quantitative measure of a bulk solids strength can only be determined after it has been consolidated to a known repeat stress condition.  

Particularly in the case of cohesive materials (i.e. those that present flow problems) where strength is proportional to the consolidation stress that has been applied.  

*Note that free-flowing materials can also show a stress history dependence (either over-consolidating or under-consolidating) due to particle shape and size distribution effects.  

A common example for free-flowing materials is the feeder over-pressure that occurs after silo loading (due to filling stress field), requiring a high initial torque to start the feeder, that rapidly diminishes to a much lower running torque as soon as the bulk solid starts to flow from the silo.  

Janssen effect

The Janssen effect, describing the stress distribution in the parallel section of a silo as presented below has an important bearing on the shape of the container used in bulk solid characterisation tests as the same phenomena will occur at any scale.  

If a liquid were stored in a vertical tube the pressure would increase linearly (hydrostatic equation) equal to the density x gravity x depth in tube and the vertical, radial and circumferential pressures would be equal.

When a bulk solid is stored in a vertical tube, the stress conditions are significantly different, the major stress is vertical due to self-weight under gravity, the minor stress is in the horizontal (radial) direction (typically 30 to 40% of the vertical stress) and is a function of the internal friction angle.

A low angle of internal friction results in a higher horizontal stress due to the greater weight of side-slip, whereas a high angle results in a lower horizontal stress.  

As bulk solids can support shear stresses, under the action of the horizontal wall normal stress, friction acting between the bulk solid the inner wall of the tube resists the self-weight and transfers a fraction of the vertical stress to an axial stress (buckling stress) in the tube.  

This wall friction effect has a negative feedback effect on the vertical stress over the depth of the tube so that typically at a depth of 3 tube diameters, the vertical stress becomes constant as the self-weight of a horizontal slice of bulk solid is supported by the wall frictional force around its circumference.  

A vertical force balance when integrated for depth gives the Janssen equation 1 below for vertical stress s_v distribution as function of depth z in the tube.  

Figure 1 firstly compares the stress distribution of a bulk solid in a tube with that of a liquid of a comparable density, showing a linear increase for the liquid and an increase at a reducing rate to constant value at great depth for the bulk solid.  

Secondly, it shows how the vertical stress distribution changes when a large vertical load is placed on the top surface of the bulk solid.  E.g. consolidation of a bulk solid in a uniaxial cell and illustrates why all shear cells have shallow depths.  

Overview of different qualitative test types

The poured and tapped bulk density test (fig 2a) comprises a glass test tube with graduations for volume, that is mounted on a platen above a rotating cam to provide taps at a controlled rate and amplitude.  

Bulk solid is poured into tube.  The fill volume and sample weight are measured to determine the poured bulk density.  The tube is returned to platen and subjected to 500-750 taps and the fill level remeasured to determine the tapped bulk density.  

*Note: if the bulk solid is cohesive with agglomerates these should be broken up by pre-screening.  

If tapping generates a significant increase in bulk density it indicates cohesion and potential for poor flowability, whereas a free-flowing material will show no change.  i.e. cohesive interparticle forces hold particles in an open packing structure following pouring until an external force “tapping” overcomes this and compacts particles into a closer packing structure.

Difference in bulk densities is generally presented as the Hausner ratio or Carr index (see equations below).  This is a good screening technique for identifying cohesive materials.  

Note however, the measured bulk density can be affected by the Janssen effect due to high height to diameter ratio of tube, particularly if material has a high friction against the glass tube.  The tapped bulk density can be used for estimating pack volumes and storage vessel capacities.  

The flow funnel test (fig 2b) comprises a flat-bottomed core-flow cup fitted with interchangeable orifice plates with outlet diameters typically in the 1-20mm range.  A known quantity of bulk solid is filled into the flow cup and the discharge time is recorded.

This is effectively a flow or no-flow test to determine whether a bulk solid will flow through a given outlet size and then measures the gravity discharge rate (grams per second) for free-flowing or easy flowing materials.  

The test will give a no-flow result for cohesive or very cohesive bulk solids due to cohesive arching.  This test can be useful when it exactly matches the process such as die filling operations.  However, care should be taken over scaling results.  

For free-flowing bulk solids, the flow rate is normally proportional to outlet size, but changes in cup shape can significantly change the result.  I.e. discharge rate from a steep wall cone is significantly lower than that from a flat-bottomed vessel of the same orifice size due to differences in the height of the dynamic arch formed over the outlet.  

The angle of repose test (fig 3a) discharges bulk solid from a funnel onto a surface to form a conical heap then measures the height and diameter to determine the angle of repose a°.  Suitable only for free-flowing materials.

Cohesive bulk solids generally will not discharge from the funnel and if a heap is formed it often displays a variable slope that is shallower at the base and steeper at the apex.  

Particularly for cohesive materials the angle of repose is not an intrinsic property of the bulk solid but is strongly influence by how the heap is formed.  

Consider that if the bulk solid drops from a significant height and becomes fluidised an angle of repose approaching 0° could be formed, while if the same material is highly consolidated an angle of repose of 90° can be formed (see “sand castle” test described later).  

The angle of repose test is useful for estimating the volume of a stockpile, or ullage in the headspace of a silo, but is not suitable for assessing flowability.

*Note: The angle of repose a° can also be measured in a rotating drum to give static angle prior to slip and the dynamic angle formed after slip.  This can be used for screening free-flowing bulk solids (as they show only a small consistent difference in static and dynamic angles) from cohesive bulk solids which show a bigger difference and more inconsistent results.  

The flow energy test (fig 3b) where bulk solid in a slender tube is first conditioned by running a rotating impeller up and down to loosen the packing structure.  

The impeller is then driven down into the bulk solid to compact it while the flow energy is determined as the area under the impeller torque versus displacement curve.  

Note that the above test sequence does not pre-consolidate the bulk solid like a conventional flowability measurement (flow function test described below see fig 4), i.e. flow energy is measured during consolidation.  

This is a dynamic flow test, so no use for assessing flow from storage equipment, but has been found useful in combination with other flow properties for correlating flow behaviour in a screw conveyor from machine learning experiments.  

Miscellaneous tests from Carrs’ flow indices such as; bulk solid build-up on a given sieve, or angle of bulk solid discharge from a spatula can be useful if they closely match your process but are not reliable measures of flowability.  

Overview of different quantitative test types

The “sand castle” or uniaxial unconfined failure test (fig 4a) is the simplest way to illustrate the concept of a bulk solid flowability measurement.  

In the first stage a cohesive bulk solid is consolidated by a known vertical stress in a cylindrical mould, the mould is then removed to reveal an unconfined bulk solid “sand castle”, i.e. the horizontal stress is zero.  

In the second stage an increasing stress is applied to the top of the “sand castle” until expansive failure occurs.  The test is then repeated using a range of increasing known consolidation stresses.  The flow function (fig 4b) is determined by plotting the consolidation s₁ stress versus the unconfined strength at failure s_c.  

If the material is free-flowing it will gain no strength and the flow function would lie on the horizontal axis.  Bulk solids of increasing cohesive strength display increasing flow function gradients and intercepts with the vertical axis.  The flow function represents the stress conditions at arch over the outlet of a silo see fig 5a.  

While the above test clearly illustrates the concept of the flow function, it is fraught with difficulties;

• firstly, the cohesive strength must be greater than the self-weight of the sample otherwise it will fail when the mould is removed,
• secondly there is the problem of minimising wall friction inside the mould which means that the actual consolidation stress at the base is significantly lower than that applied (see fig 1 red line), and so the measured unconfined strength is proportionately lower.  

The uniaxial shear cell test operates in a similar manner (see fig 5b), but the shallow proportions eliminate the wall friction effect inherent in the “sand castle” test.  However, the uniaxial consolidation process still results in a lower strength by comparison with the shearing method employed in the shear cell below.  

The shear test is the preferred method for flow function measurement using a shear cell (see fig 6a) – commercially available automated testers include the; Brookfield PFT, Schulze RST, Freeman FT4 shear cell, as well as manual Jenike shear cell.  

The shear cell is an indirect test i.e. it determines each point on the flow function from an extrapolation of a locus of failure points see fig 6b.  

Prior to each failure point measurement, the bulk solid is pre-sheared to a critical state “bulk density” (normal stress of s_E, shear stress of t_E), this gives a repeatable starting consolidation stress that is representative of the shear process in a storage vessel during flow and maximises the unconfined failure strength at failure.  

Mohr stress circles are fitted tangential to the measured failure locus to determine the consolidation stress s₁ and unconfined failure strength s_c, for plotting the flow function.  The shear cell also determines the effective internal friction angle d.  

These measurements can be used to assess flowability or be used with the below wall friction and compaction tests to design a silo for reliable gravity flow or to determine the silo actions for structural design.  

The wall friction test using a shear cell (see fig 7a) consolidates the material against the wall to a given normal stress and then measures the wall shear stress over a range of reducing wall normal stresses to create a wall friction locus, from which the wall friction angle j_w is determined from the gradient.  

This represents the limiting gravity chute angle for flow down the wall. This can be used to calculate the limiting mass-flow hopper half angle, or resistance to flow in mechanical conveyors/feeders for power and torque calculations.  

The compaction test (fig 7b) bulk solid is poured into a cell of known volume and weighted, the change in volume (bed height) is measured as a function of increasing consolidation stress applied to the lid.  

The low height to depth ratio of the cell eliminates the Janssen effect.  The bulk density can be used to determine pack or storage vessel volumes and feed rates of mechanical conveyors.  

Summary

There are a wide range of bulk solid characterisation tests.  The qualitative tests are simple tests that maybe useful for screening free-flowing bulk solids from cohesive, or for process characterisation if the test closely matches the process.

Quantitative tests measure fundamental bulk properties that can be used for flowability ranking, silo geometric design for reliable gravity flow, determining silo actions and design of mechanical and gravity conveyors.