Yield stress hoppers

A powder s strength increases significantly with increasing previous compaction. The relationship between the unconfined yield stress/, or a powder s strength, and compaction pressure is described by the powder s flow function FE The flow function is the paramount characterization of powder strength and flow properties, and it is calculated from the yield loci determined from shear cell measurements. [Jenike, Storage and Flow of Solids, Univ. of Utah, Eng. Exp. Station Bulletin, no. 123, November (1964). See also Sec. 21 on storage bins, silos, and hoppers.]... 

The unconfined yield stress is a measure of the stress necessary to cause a material unsupported in two directions to fail in shear. This is what must happen when an arch fails within the powder or at the hopper opening. The effective... 

Considering the equilibrium of a stable arch, show that the minimum diameter of the orifice in a conical hopper, d,ma, which is defined to account for the arching, can be related to the unconfined yield stress of the bulk material by the relation... 

This may be plotted on the same axes as the powder flow function (unconfined yield stress, Gj and compacting stress, gq) in order to reveal the conditions under which flow will occur for this powder in the hopper. The limiting condition gives a straight line of slope 1 /ff. Figure 10.7 shows such a plot. 

The powder flow function which can be represented by the relationship, ay = a, where Oy is unconfined yield stress (kN/m ) and gq is consolidating stress (kN/m ) Determine (a) the maximum semi-included angle of a conical mild steel hopper that will confidently ensure mass flow, and (b) the minimum diameter of circular outlet to ensure flow when the outlet is opened. 

The axisymmetric nature of conical hoppers results in es = 0 and, according to Eq. (2.20), cre = (compatibility requirement, i.e., the relationship of strains. This relation, with the aid of constitutive relations between stress and strain (e.g., Hooke s law), provides an additional equation for stress so that the problem can be closed. However, the compatibility relation for a continuum solid may not be extendable to the cases of powders. Thus, additional assumptions or models are needed to yield another equation for stresses in powders. 

It is required to design a mass flow conical hopper with the volume capacity of 100 m3 to store a cohesionless material of bulk density 1,700 kg/m3 and an angle of internal friction 40°. Four sets of shear tests have been conducted on the material and results for the unconfined yield strength and the corresponding consolidating stress are as follows ... 

In a state of incipient failure, the yield locus is tangent to the Mohr circle. The Mohr circle graphically represents the equilibrium stress condition at a particular point at any orientation for a system in a condition of static equilibrium in a two-dimensional stress field. The equilibrium static conditions can also be applied to sufficiently slow steady flows. The maximum principal stress in Fig. 6.4(b) is called the unconfined yield strength. This is the maximum normal stress, under incipient failure conditions, at a point where the other principal stress becomes zero. Such a situation occurs on the exposed surface of an arch or dome in a feed hopper at the moment of failure see Fig. 7.5(b). In the analysis of bridging in feed hoppers, the unconfined yield strength becomes a very important parameter. The magnitude of the unconfined yield strength is determined by the YL and depends, therefore, on the consolidation pressure and time. 

The stress acting on an exposed surface is also the only non-zero principal stress, because the exposed surface is assumed self-supporting and traction-free (i.e., no shear stresses acting on the surface). The flow factor is determined by the geometry of the hopper and the properties of the bulk material. Another function used by Jenike is the flow function . This flow function is the ratio of the consolidating pressure CTi to the unconfined yield strength as defined in Section 6.1.2 ... 

The kinematic angle of friction between powder and hopper wall is otherwise known as the angle of wall friction, This gives us the relationship between normal stress acting between powder and wall and the shear stress under flow conditions. To determine it is necessary to first construct the wall yield locus... 

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. 

The choice given to the production engineer or research powder technologist is thus dependent upon the level of information required. In the design of mass flow hoppers there tends to be a greater need for knowledge on yield loci, stresses and failure criteria than that possibly needed for routine powder quality control. 

The anisotropic behaviour of bulk solids mentioned in connection with Fig. 5 (procedure III) is of no influence in the design of silos for flow. With help of Fig. 5 and 6 it was explained that steady state flow was achieved with Oi (at steady state flow) acting in x-direction. The unconfmed yield strength was also measured with the major principal stress acting in x-direction. During steady state flow in a hopper the major principal stress is in the hopper-axis horizontal. In a stable dome above the aperture the unconfined yield strength also acts horizontally in the hopper axis. Therefore, the Flow Function reflects reality in the hopper area. 

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. 

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