Stresses in Bins and Hoppers, by Gabriel I. Tardos

-- 6. Arching in Hoppers:
Jenike's Method of Hopper Design --

A typical stress distribution in a silo filled with a bulk powder is shown in Fig. 7 and, as seen, the material in the bin is subjected to increasing compression as it moves down from the top of the fill. In the converging hopper however, the material is under decreasing stress and becomes fully un-compressed at the apex. It is entirely possible therefore that, if a sufficiently small hole is cut out of the lower part of the hopper, the material will not flow out since the stresses around the apex are quite small. This is in marked difference to the behavior of fluids where the pressure increases continuously (following the hydrostatic stress distribution y,h in Fig. 7) and reaches a maximum at the bottom. To discharge powder from the silo, providing a large enough opening at the bottom of the hopper to ensure flow is necessary. Jenike's method of hopper design is based on this observation and on the fact that the material forms arches that bridge the opening and that have to be broken continuously to achieve flow. Hopper design is reduced in this context to the calculation of the minimum outlet dimension B (see Figs. 1a and 9) for which arches of material bridging the exit will break under their own weight. This dimension will depend on the powder properties that determine the stress in the hopper and the hopper half-angle, .

Fig. 9 - Plane arch in a hopper

Fig. 9 shows a plane 2D arch formed in a hopper. The maximum stress in the arch (due to its weight) is seen to be proportional to its span B and to the material's bulk density; the location of this maximum stress is at the abutment of the arch at point L and can be approximated as [23a]      1,a B = 2 r sin The above equation can be generalized [Shamlou, 1990] for a plane and a 3D conical arch to read [23b]      1,a = [2 r sin / m] where m = 1 for the 2D and m = 2 for the conical hopper and B = 2 r sin . To find the condition of collapse of the arch in the hopper, an additional important powder bulk property has to be measured experimentally. This is the so called unconfined yield strength that is the maximum normal stress under which a powder with a free, unstressed surface will yield or deform indefinitely. Fig. 10 shows the simplest procedure to measure this powder property.

Fig. 10 - Measurement of the unconfined yield strength

The specimen is contained in a cylindrical vessel and pre-compressed or consolidated by the normal stress, 1, as shown. In a subsequent step, the stress is halted and the walls of the container are removed exposing the free powder surface. The specimen is compressed again until it gives way under the normal stress denoted in Fig. 10 as fc. The experiment is repeated several times at ever-increasing consolidations 1, and the values of the unconfined yield strength are measured. Pairs of stresses are correlated in the form [24]      fc = FF(1) This experimentally obtained relationship is an intrinsic characteristic of the powder just like its bulk density, B and its angle of internal friction, , and was denoted by Jenike as the Flow Function. In industrial practice, the above powder properties and the angle of wall friction, W, are measured in a special device invented by Jenike and called a split cell. As seen in the above correlation, the yield strength, fc, of the material is a direct function of the consolidating stress, 1, in that larger strength is associated with higher consolidation. This result has a major implication in silo design in view of the variation of consolidation stresses in the silo as calculated in the previous sections and as shown in Fig. 7; the material strengthens as it moves downward in the bin but subsequently becomes weaker inside the hopper as it approaches the apex. The majority of fine powders exhibit an unconfined yield strength, fc, when subjected to consolidation; these are usually called cohesive materials. There is however, a class of larger size powders for which the value of fc is practically zero for even large consolidations. These are the so-called non-cohesive or free-flowing materials, a few examples of which are given in Table 1. The most commonly known free-flowing material is fine, dry sand. Table 2 shows flow functions FF, for several cohesive materials as well as values of the angle of internal and wall friction, as measured by Drescher et al. (1995). The design criteria for hoppers can now be stated qualitatively using the maximum stress at the abutment of the arch, 1,a, and the unconfined yield strength, fc, as the so-called flow-no-flow criteria. For the arch to break and hence for the powder to flow, the maximum stress has to overcome the yield strength [25]      1,a fc The critical condition occurs when the equality holds and where the critical span, Bcrit, is obtained. The solution to the above problem comes down to finding the position, r, (see Fig. 9) or equivalently the span, B, where Eq. 25 holds with 1,a and fc given by [26]      1,a = [2 r sin ] / m = B / m ;       fc = FF(1) and where 1 is the consolidating stress in the hopper. This stress has been calculated previously and is given by Eqs. 15, 16 or 17. One has to note that the stresses y and r in Eqs. 15 and 16 are not principal stresses and cannot be used directly in the above equation (they can however be easily transformed to yield principal stresses as seen in the Appendix). The stress 1 in the convex slice given in Eq. 17 is, however, a principal stress -- there are no shearing stresses in the slice -- and can be combined with the above equations to close the system [27]      |1| = K F r / (1 + m G) Eqs. 26 and 27 form, with the condition in Eq. 25, a system that is quite straightforward to solve for the radius r (or span B). The difficulty arises in the non-explicit form of the Flow Function, FF, which is usually given graphically. To overcome this difficulty, Jenike proposed to use the special dependence of both Eqs. 23b and 27 on the first power of the radius, r, and to calculate the ratio [28]      ff = 1 / 1,a = K F m / [2 sin (1 + m G)] It is seen that this ratio depends only on material properties and not on position and was denoted by Jenike as the flow factor. In view of this, the relations in Eq. 26 become [29]      fc = FF(1) ;       1,a = (1 / ff) 1 and can be solved graphically by intersecting the curve of the Flow Function FF with a line through the origin with the slope, tan = 1 / ff, in a coordinate system with 1 as the ordinate as depicted in Fig. 11.

Fig. 11 - Graphic solution of the design equation

Below the line, continuous collapse of the arch ensures flow while above it arcing occurs. At the intersection the condition in Eq. 25 is satisfied and the critical span is obtained from Eq. 26 as [30]      Bcrit = m 1,a,crit / = m fc,crit / where the critical values are taken as shown in the figure. Since the Flow Function, FF, is a measured quantity, the only challenge in the design procedure is the calculation of the flow factor, ff. Two flow functions are given in Fig. 11 one for material consolidated for a short time called the instantaneous flow function and another denoted the time-flow function, obtained by consolidating the material for a long time. In the example given above the instantaneous values were used yielding a critical span given by Eq. 30. The time flow function gives a somewhat larger critical value for the yield strength, fc, and therefore a larger value for the span, Bcrit. The smaller span is used for a silo that is in continuous use and where material is loaded and unloaded with very little storage time. The larger value is employed for silos where material is stored for long times. The time flow function has to be obtained experimentally for consolidation times equal or longer than the typical storage time. For the case of free-flowing materials such as those presented in Table 1, the unconfined yield strength is fc = 0 and all point fall below the line in Fig. 11. This ensures that such a material will, at least in principle, flow out through any opening with a span B larger than the grain size, hence the name for these powders. Table 1.a - Physical properties of several free-flowing powdery materials Table 1.b - Interaction properties of several free-flowing powdery materials In the case of cohesive materials the Flow Function, fc = FF(1), may be computed from a Warren-Spring equation as: [31]      fc = E [(1 / F + 1)1/q - 1] where E, F, and q are material constants given in Table 2b. Table 2a - Physical properties for several cohesive materials [after Drescher et al., 1995] Table 2b - Flow Functions for several cohesive materials [after Drescher et al., 1995] An alternate way of expressing the Flow Function is by use of a linear fit: [32]      fc = K 1 + L where values for constants K and L are given in Table 2b.