Failure function powders

When failure does occur, the flow is frictional in nature and often is a weak function of strain rate, depending instead on shear strain. Prior to failure, the powder behaves as an elastic solid. In this sense, bulk powders do not nave a viscosity in the bulk state. 

A precise definition of the flowability of a powder is only possible with several numbers and curves, derived from a family of yield loci of the powder (measured with a shear cell) - see section 4 for further detail. Jenike23 proposed a simpler classification, according to the position of one point of the failure function (at a fixed value of the unconfined yield strength, say 5 lbf (22.3 N) with the Jenike shear cell, i.e. 3112 Pa or 65 lbf/ft2) with respect to the flow factor line (straight line through the origin, at a slope l///where//is the flow factor) - see Fig. 8 for a schematic representation of this. 

In addition to the above classification, Williams24 has also defined simple powders as those whose failure function is a straight line through the origin when designing hoppers for such powders,... 

In hopper design, the failure function which represents the strength of the powder on a free surface at different states of consolidation, is compared with another curve which describes the actual stresses in a hopper and the size of the opening is derived from this comparison to give flow every time the outlet is opened. 

J.C. Williams, A.H. Birks and D. Bhattacharya, The direct measurement of the failure function of a cohesive powder, Powder Technology, 4 (1970/71), 328-37. 

A graph of a failure function of a powder is given in Figure 2.17. Although the results of this method can be used for monitoring or for comparison. 

Figure 1.6 shows the correlation between the complex Jenike failure function ff with the simple FR (Table 1.4). These results indicate that for values of FR less than 1.25, the industrial powders tested were free flowing. Above an FR value of 1.55, the powders could be classifled as cohesive with intermediate degrees of flowability between FR values of 1.25 and 1.55 as illustrated in Table 1.5. 

Jenike failure function (Jf) This is the reiationship between the unconfined yield strength (fa) and the major consoiidation stress (cti ). A plot of the values fa versus cti shows the possible relationship of the rate of flow of powdered material out of hopper orifices. Jenike could classify the flowability of powders from selected ratios of these values. 

If the failure function does pass through the origin then these powders are defined as simple powders (Williams Birks 1965, 1967). 

Jenike failure function Description of bulk powder flow... 

The shear flowability index, n, was found, from past observations (Farley Valentin 1965, 67/68), to be independent of the bulk density of sheared compacted powder. Because of this independence of particle size from bulk density it is now realised that the shear flowability index, n, from the Warren Spring equation and the Jenike internal angle of friction may be the preferred parameters to eharacterise and quantify the flowability of powders. Jenike and others (Williams et al. 1970/71 Williams Birks 1965 Hill Wu 1996 Cox Hill 2004) selected the Jenike failure function to be one of the best indicators to predict the ease of powder movement and powder flowability. 

Kurz and Minz (1975) investigated the flowability of powders in terms of the relationship between the unconfined yield strength, /c, and the major consolidation, tri - the failure function of Jenike — for different-sized limestone powders (3.1-55.0 um) having either a narrow or a wide particle size distribution. The width of the distribution was defined by a variation coefficient, Cv, where Cv = cTstatAi.3, with agtat as the standard deviation of the particle size profile and X1.3 the average particle size of the number-volume diameter distribution of the limestone particles. A narrow distribution was considered to have a Cv < 0.5 while a wide distribution had a Cv > 0.5. 

Table 1.15, taken from the work of Schulze (1995, 1996a,b), indicates a wide range of apparati and protocols to measure a powder yield strength or a (Jenike) failure function for use in the determination of bulk powder flowability. 

The second objective is to examine how different functions of the equilibrium stress for the failure property coefficients, when inserted in the simple powder equation, affect the form of the failure function and explore the implication for sizing hopper outlets. 

When oe becomes zero, q = ks, therefore (oe + T) can never reach zero and the failure function will not pass through zero, but instead only approaches the f = oie line which instantaneous functions can never intersect. This type of property is, of course, describing wet powders, namely, those whose moisture is greater than relative humidity saturation but less than void saturation. Powders with moisture less than relative humidity saturation behave as dry powders, while a truly dry powder, in the chemical sense, exists only in very special conditions and industrially is an extreme rarity. 

In 1967, Williams and Birks, showed that it was possible for a simple powder to possess a linear failure function which also passes through the origin[5]. Such a failure function can not give a intersection with a line whose slope is equal to the reciprocal of the flow factor (part of the design procedure) and also passes through zero. Should the failure function lie above the flow factor line, the powder will not flow out of the channel while, if lying below, it will flow out of an infinity small hole, according to the theory. Clearly this latter statement needs further examination. 

If a particle of the powder is larger than the outlet it will not flow out. If the outlet is several-fold larger than the particles, the powder may not flow out. If the outlet is manyfold larger than micrometer sized particles, the powder may not flow out. Obviously, extrapolating the failure function down to the region of extremely low stresses is not tenable because one moves into a regime where other mechanisms, in addition those of the Jenike theory, start to become significant. 

While the above examination considered the implications of the linear failure function of a simple powder, it is abundantly clear that the same arguments apply to any non-linear failure function passing through the origin. Hence, any dry hard-particle powder will flow out of a small outlet subject to the intervening very low-stress mechanisms. 

Within the limits of the assumptions, used to derive equation 3, the mathematical conditions for powders to have a failure function passing through zero have been demonstrated. There is a very strong possibility that any air-dried fine mineral powder will possess a failure function which passes through zero. The difference in dealing with stress on an applied stress basis and a compound stress basis has been explained. 

Figure 21.9 Failure diameter of TNT as a function of initial density (Ref. 7). I. Pressed or powder Grains ize (I) 0.01-0.05 mm (b) 0.07-0.2 mm. 2. Cast (a) Poured clear (b) creamed (c) creamed with 10% powder added.

Powders can withstand stress without flowing, in contrast to most liquids. The strength or yield stress of this powder is a function of previous compaction, and is not unique, but depends on stress ap ication. Powders fail only under applied shear stress, and not isotropic load, although they do compress. For a given apphed horizontal load, failure can occur by either raising or lowering die normal stress, and two possible values of failure shear stress are obtained (active versus passive failure). 

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