Powder flow function and numerical characterization of powder flowability

The ratio of c to D is termed the Jenike hopper flow factor, ff; see Equation 3.3. A high value of

ff represents poor flowability because a high value of c means greater consolidation and a low value of D indicates the possibility of arching is high (Jenike, 1964). Recalling the definitions of

c, D, and y in Section 3.2.1 and Figure 3.2, D has to be greater than y for powder failure or flow to happen; therefore a flow–no flow criterion in the form of Equation 3.4 is obtained; y is a function of c and the plot of experimental y against c gives a graphical representation known as

powder flow function, FF.

Hopper flow factor,

ff =c

D

(3.3)

For powder flow,

c

ff >y (3.4)

The limiting FF values or conditions for flow can be determined with Equation 3.5, which is a straight line with a slope of 1/ff. Based on this, Jenike (1964) suggested four divisions or classifications of powder flow, namely very cohesive or non-flowing, cohesive, easy flowing, and free flowing, see Table 3.1; these divisions are arbitrary and they are known as Jenike’s criteria for powder flowability. The limiting FF values and criteria for flowability can be superimposed on the c:y plot to give the boundaries for transition in powder flow and hence qualitative flow information; an example is given in Figure 3.3.

Limiting condition for flow,

c

Table 3.1 Jenike’s limiting flow function values and arbitrary powder flow divisions (Jenike, 1964)

Jenike’s Limiting FF Values Arbitrary Powder Flow Divisions

FF < 2 Very cohesive and non-flowing

2 < FF < 4 Cohesive

4 < FF < 10 Easy-flowing

10 < FF Free-flowing

There are two typical flow functions in Figure 3.3; Line A is for a free flowing powder and Line B is for a cohesive powder. For the free flowing powder, the flow function is generally constant and its flow behaviour is unaffected by c. For the cohesive powder, the flow function increases with c and is nonlinear, as demonstrated with Line B; this trend is common for most powders that show a certain degree of cohesiveness, see for example Kurz and Münz (1975). With reference to Line B, the cohesive powder tends to flow better under high consolidation stresses and its flow becomes poor at low stress levels.

Figure 3.3 Examples of powder flow functions with real data and Jenike’s criteria for powder flowability

The utility of classifying powder flowability according to Jenike’s criteria has been commonly used, see for example Stanley-Wood et al. (1993), Teunou et al. (1999), and Vasilenko, Glasser and Muzzio (2011). However, there are two shortcomings in Jenike’s classification.

First, it should be noted that Jenike had conveniently forgone some precision in his analysis in order to achieve brevity in powder flow definition, see Jenike (1964); therefore caution should be exercised in the use of the classification. It is also worth noting that the original

intent of his work was for the design of storage hoppers and silos, and not the characterization of powder flowability.

Second, it is not possible to completely describe powder flowability with only one numerical value on the c:y plot because powder flow function changes with increasing consolidation stress; the general exception is powders that are free flowing. Therefore, the c/y at a particular consolidation stress cannot be used to infer flowability at other stress levels. For accurate flow characterization, several numbers and flow function curves are required (Jenike, 1964; Schulze, 2008).

3.2.3 Cohesion

Cohesion, C, is the shear stress of a consolidated powder when no normal stress is applied to the shear plane; it is related to the interparticle forces that must be overcome before powder failure or flow commences (Schulze, 2008). With reference to Figure 3.1, C is obtained by extrapolating the yield locus to the -axis of the : plot. Besides powder flow function, C is an important powder and flow property; the following lists three key investigations on the C of fine powders that have prompted the interest in C in this work.

Orband and Geldart (1997) investigated the relationships between the C of lactose powders and soda ash and d32. C was measured at consolidation stresses below 15 kPa and in the unconsolidated state with a torsional device that operated on a principle similar to that of an annular shear cell; the apparatus measured C directly and eliminated the yield locus extrapolation step. The plot of C against d32 for each powder revealed a critical particle diameter; the critical diameter was 52–60 μm for lactose powders and 50 μm for soda ash. Below the critical diameter,

C increased progressively with decreasing particle diameter and was a function of consolidation stress. Above the critical diameter, constant values of C that fluctuated between 0.1 kPa and 0.2 kPa were observed; the powders were “free flowing” and the influence of consolidation stress was insignificant. Orband and Geldart further noted the dependence of C on the reciprocal of particle diameter and pre, but made no proposal on any correlation that could simultaneously relate C to both factors.

Vasilenko et al. (2011) measured the C of selected pharmaceutical blends at 3–15 kPa with a rotational shear cell and correlated the data with the Flow Index determined with the Gravitational Displacement Rheometer, GDR; the Flow Index is an indicator of powder avalanche activity measured during the rotation of the drum of the GDR. A linear relationship between C and the Flow Index was observed, and the slope of the straight line changed with the composition of the pharmaceutical blends. In a later report, Vasilenko, Koynov, Glasser and Muzzio (2013) observed similar and consistent findings; their measurements were done with selected catalyst powders and over a lower consolidation stress range, 0.5–3 kPa.

The approach used by Vasilenko and colleagues is empirical, and they have demonstrated correlations between C, which is a powder flow property measured under consolidated and confined conditions, and the GDR Flow Index, which is a measure of powder flowability under unconfined conditions, see Faqih, Chaudhuri, Alexander, et al. (2006) and Faqih, Chaudhuri, Muzzio, et al. (2006). However, there has not been evaluation and discussion on the correlation between the GDR Flow Index and cohesion under zero consolidation stress, C0, which is the stress of unconsolidated powders that has to be overcome for flow to initiate under unconfined conditions; there is therefore scope for further investigation.

In this chapter, the C and C0 of selected powders are determined and their relationships with particle diameter and pre are investigated. Correlations between C0 and the GDR Flow Index are assessed and discussed in Chapter 6, which is on powder tumbling.