The Hopper Flow Factor, ff

The hopper flow factor, ff, relates the stress developed in a particulate solid with the compacting stress acting in a particular hopper. The hopper flow factor is defined as  

A high value of ff means low flowability since high ac means greater compaction, and a low value of cd means more chance of an arch forming. 

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H 9 ) is a dimensionless function derived from first principles and is given by Figure 7 [for the complete derivation of H 0 ), which is beyond the scope of this chapter, see Ref. 4]. /ent, with units of force/area, is the unconfined yield strength at the intersection of the hopper flow factor (ff, a derived... 

A useful approximation of B for a conical hopper is B = 22f/a, where a is the bulk density of the stored product. The apparatus for determining the properties of solids has been developed and is offered for sale by the consulting firm of Jenike and Johansen, Winchester, Massachusetts, which also performs these tests on a contract basis. The flow-factor FF tester, a constant-rate-of-strain, direct-shear-type machine, gives the locus of points for the FF cui ve as well as ( ), the... 

Fig. 9 shows the unconfined yield strength for various vibration velocities versus the major principle stress during consolidation oi. The dashed lines show the range of effective wall stresses 0 = 0]/ff of a cohesive powder arch for common values of the flow factor ff. The intersection point of Oc and oT delivers the so-called critical unconfined yield strength ac.crii-Eventually, the minimum outlet diameter to avoid bridging in a mass flow hopper bmin is directly proportional to Cc,crit- In the example, shown in Fig. 9, the critical unconfined yield strength (and hence bmin) can be strongly reduced in presence of vibrations. 

The doming - no doming criterion was explained qualitatively with help of Fig. 7. For predicting the critical height h quantitatively the dependencies Oc = f(cTi) and 0 = f(oi) have to be known. Since doming is mainly a problem in the hopper section, only this part of the silo has to be considered, 0 and Oi can be calculated, a and 0 both increase linearly with distance from tire hopper apex and both are equal to zero at the apex. Therefore, the ratio of both, called flow factor ff = oi / cti, is a constant in the hopper, ff depends on the parameters (pe, cpx and 0, already used for the mass flow/funnel flow-decision in Fig. 5, and can be read from equivalent graphs in the literature [1]. The dependence o, = f(oi) is a bulk solid property, called flow function. It can be measured with help of shear testers (see chapter 5). 

Flow factor graph (similar to Figure 3.11) showing the value of ff and the hopper half angle. 

In order to obtain quantitative results, the flow factor has to be determined this requires knowledge of the stress field in the hopper. Closed-form expressions for ff are not available, except for the simplest case of flow through a straight cylinder. Results from numerical analysis are given by Jenike [8, 9] in graphical form for plane symmetry and axial symmetry. 

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