1.5.1 Poured and drained angles
There is much ambiguity when the term ‘angle of repose’ is used, because the angle of internal friction and angle of slide have, at times, been referred to as ‘an angle of repose’. The angle of repose is commonly defined as the angle between the free surface of a pile of powder on a horizontal board to the horizontal plane. This angle is, however, dependent upon the method of formation of the pile of powder. The test is only relevant to slightly cohesive powders or non-cohesive powders, which can form a specific angle when either poured or drained. The poured angle of repose is the angle measured from a pile poured freely onto a flat surface.
The drained angle of repose is the angle measured on or from the conical surface of powder in a flat-bottomed container after the powder has been discharged or drained from the container via an orifice.
These two values are different, the latter being greater than the former, because the poured angle has particles sliding and rolling down the sloped powder surfaces causing separation, whilst the drained angle tends to achieve a convergence or mixing of particles within the remaining piled up material nesting in the container.
1.5.2 Static angle of repose of a heap
Angles of repose are measured in various ways. The angle of repose was defined by von Terzaghi (1925, 1943; Peck & Gholamriza 1996; von Terzaghi & Peck 1967) as the angle between the horizontal and the slope of a heap of soil dropped from a known elevation. The angle depends, however, on the diameter of the pile, higher slopes resulting from small heap diameters. With large diameter heaps segregation by sliding can also occur. Figure 1.15 shows a simple angle of repose measurement apparatus, while Figure 1.9 shows an apparatus designed by Carr which incorporates a protractor, an indicator wire and a jarring device to measure the angle of fall. The lower the angle of repose the more flowable a powder becomes because the angle of repose is a direct indication of the potential flowability of a powder. This technique is a relatively simple method and may be used, at times, to measure indirectly particle properties affecting flow such as:
r
Shape and sizer
Surface arear
PorosityFunnel
Bridge
5.2 cm
Figure 1.15 Simple apparatus for the measurement of angle of repose.
r
Fluidityr
Bulk densityAlthough the angle of repose has been used to design feeders and storage bins, there is, nowadays, more sophisticated and accurate instrumentation available to measure other bulk powder factors such as:
r
Jenike angle of frictionr
Wall frictionr
Cohesionr
Tensile strengthwhich are generally better criteria for use in mass flow hopper design.
Probably the only general powder standard to describe the procedure to measure an angle of repose is that described by British Standard (BS 4140-9: 1986) while there are three ISO standards for different powdered materials, ISO 902: 1976, for alumina; ISO 4324: 1977, for powders and granules and ISO 8398: 1989, for fertilisers. The apparatus consists of a glass or polythene funnel with a cut-off stem mounted on a tripod at a known height (5.2 cm) above a plate of metal or wood. This bottom plate has a series of concentric circles inscribed upon its surface to indicate the different angles of the poured conical surface. The diameters of the circles are calibrated in degrees with a step size of 2◦ (Figure 1.15). A commercial instrument for angle of repose measurement for use in the process control of semi-cohesive powders has been developed (Powder Research Ltd, Harrogate, UK) from
Table 1.10 Flowability indicators and categories of powder flow from USP 29-NF24 (2006).
Description of flow Angle of repose (degree) Carr’s compressibility (%) Hausner ratio (−)
Excellent/very free flow 25–30 <10 1.00–1.11
Good/free flow 31–35 11–15 1.12–1.18
Fair 36–40 16–20 1.19–1.25
Passable 41–45 21–25 1.26–1.34
Poor/cohesive 46–55 26–31 1.35–1.45
Very poor/very cohesive 56–65 32–37 1.46–1.59
Very very poor approx non-flow >66 >38 >1.60
the work of Wouters and Geldart (1996). The magnitude of the angle of the heap can be used to quantify, in a similar fashion to the Carr’s compressibility percentage, Jenike failure function and Hausner ratio, the ease of powder flow (Table 1.10).
1.5.3 Angle of fall
To measure the relative flow and stability of dry material, Carr dropped a small weight (approx. 100 g) at a distance of 17.8 cm (7 in) onto the horizontal surface upon which a heap of powder had been built in a manner akin to the determination of an angle of repose. Free-flowing material will slump or fall down to assume a different geometry from the original pile of powder – termed the angle of fall or slump. A fluid material, which can cause flooding, will, however, collapse under the impact of the weighted steel bush on to the horizontal surface, expelling the entrapped air. The lower the angle of fall the more floodable the material (Figure 1.9).
1.5.4 Angle of difference
The difference between the angle of repose and the angle of fall gives an indication of the potential floodability of materials. The greater the angle of difference the better the flowability, fluidisability and floodability.
1.5.5 Dynamic angle of repose
One of the adjuncts to angle of repose measurement is the ‘slump test’ (see Section 1.5.3) which simulates the movement of particles in a heap over each other and which can indicate a degree of dynamic cohesion or opposition to shear within a bulk powder. Following from the work of Kaye et al. (1995, 1997) on the sliding of particles over powder surfaces, the Aero-Flow Powder Flowability Analyzer (TSI Instruments Ltd) was developed. This dynamic angle of repose instrument uses a slowly rotating drum. Eventually, as the powder rotates within this transparent disk, an unstable condition is reached in which particles slide down the bulk surface of the powder under the influence of gravity to create an avalanche. A photoelectric detector monitors this avalanching behaviour to produce a series of ‘saw- tooth’ plots termed the ‘phase space attractor map’. The periodicity, in terms of the time taken to achieve an avalanche, and shape of these plots, termed the ‘strange attractor’, provide information on powder flowability. The more scattered the data from the strange attractor the more cohesive the powder. Lee et al. (2000) used the mean time to avalanche as the criterion to quantify the flowability of six pharmaceutical excipients which were subsequently compared with other laboratory-derived powder flow indicators.
Table 1.11 Flow indicators for six pharmaceutical powders.
Static angle Rank order from dynamic
Powder of repose Hausner ratio Carr’s (%) angle of repose
Aspartame 52 1.92 48 1 cohesive
ML001 50 1.54 35 2 cohesive
Lactose 46 1.43 30 3 passable
Flour 43 1.35 33 4 fair
HPMC 33 1.32 24 5 free flow
Placebo 30 1.16 14 6 very free flow
Bodhmage (2006) following the work of Riley et al. (1978) correlated the physical prop- erties and flowability indicators of six pharmaceutical powders. The six pharmaceutical powders were Aspartame (a non-carbohydrate sweetner), Respiose ML001 (an inhalation lactose), alpha-D-lactose monohydrate, Methocel HPMC (a water-soluble hydroxypropyl
methycellulose), pastry flour and a granulated placebo mixture. The flowability indicators chosen were the static angle of repose, Hausner ratio, Carr’s compressibility percentage or flow index and the dynamic angle of repose. The dynamic angle of repose was measured by a modified Aero-Flow Analyser. When the ‘non-dynamic angle of repose’ indicators were compared with the ‘saw-tooth’ frequency of avalanching data – mean time to avalanche – from Bodhmage’s newly modified electrical capacitance tomographic instrument, it was found that all six pharmaceutical powders followed the same rank order as the more con- ventional indicators (Table 1.11).
Many of the empirical flow indicators currently used have been derived from the work of Carr (1965b,c). These diverse test methods have now been collected together in an American Society for Testing and Materials – ASTM D 6393-99: 1999 – standard which covers the apparatus and procedures for measuring the properties of free flowing and moderately cohesive bulk solids and which are collectively known as ‘Carr indices’.
Test A – Measurement of Carr angle of repose Test B – Measurement of Carr angle of fall Test C – Calculation of Carr angle of difference Test D – Measurement of Carr loose bulk density Test E – Measurement of Carr packed bulk density Test F – Calculation of Carr compressibility Test G – Measurement of Carr cohesion Test H – Measurement of Carr uniformity Test I – Measurement of Carr angle of spatula Test J – Measurement of Carr dispersibility