Stresses in Bins and Hoppers, by Gabriel I. Tardos

-- 5. Values of the Janssen Constant K --

Within the context of the theory of slices the Janssen constant K is a powder and wall property that cannot be computed from the simplified theoretical approach presented above. It has to be either measured experimentally or calculated using a more basic, detailed model of the bulk material by postulating, for example, a mode of material failure, i.e., a yield condition. This implies, at a minimum, the definition and measurement of the material's angle of internal friction, . The theoretical considerations containing such a model are beyond the scope of this presentation; here we limit ourselves to some well-known correlations to calculate the constant K, obtained either experimentally or theoretically [see Drescher 1991] and use these to predict stresses. There is a large literature dedicated to calculate Janssen's constant, and some authors use a slightly different definition [18]      K = k tan W where k is also denoted sometimes as Janssen's constant. The reader is cautioned to ascertain which of the two definitions is employed before applying the stress correlations in this section when using other published materials on the subject. In addition, different values of, K, have to be used for bins and hoppers, respectively, and some are given below. All these correlations are approximate in some way and give slightly different values even for the same conditions. Janssen's constant for bins reads [Drescher 1991] [19]      K = [(1 - sin ) / (1 + sin )] tan W where = 1 for the active and = -1 for the passive stress. For hoppers different values of the coefficient K are obtained for different shapes of the slice; for example, in the case of the plane slice one gets [20]      K = [(1 - sin cos 2 W) / (1 + sin )] tan W Considering the convex slice for which the stress is given in Eq. 17, the value of the constant K is given by [21]      K = [(1 + sin ) / (1 - sin )] and the angle of inclination of the slice , (see Fig. 6b) becomes [22]      = (1 + ) / 4 - (arcsin[sin W / sin ] - W) / 2 + where again = 1 for the active and = -1 for the passive stress. The above correlations for the plane slice in both the bin and the hopper are used in Fig. 7 to represent the average vertical stress in a silo.

Fig. 7 - Average vertical stress in a bin/hopper geometry (equations shown are for plane slices)

As seen, the origin of the axes is taken at the top of the fill in the bin and at the apex of the hopper, respectively. The joining piece between the bin and hopper is cut out in the figure since none of the equations for stress calculation are strictly valid for that region. The vertical stress in the bin increases linearly following the hydrostatic pressure at the top of the fill but deviates, as seen, to reaches at large depth, the maximum value shown in the figure. The vertical stress in the hopper is linear with the distance from the apex in the lower part of the device and deviates exponentially at larger distances away from the bottom. Equations used for calculations in Fig. 7 were Eq. 7 for the bin and Eq. 15 for the hopper with values of the Janssen constant taken from Eqs. 19 and 20, respectively. More rigorous expressions for the average stress in the hopper are given by Eqs. 16 and 17 for the cylindrical and convex slices with K given in Eq. 21. Fig. 8 shows measured and calculated average vertical and wall stresses during steady discharge from a silo.

Fig. 8 - Experimental and theoretical average vertical and normal stresses
during silo discharge [after Shamlou, 1990, data by Walker, 1966]

The active state of stress is assumed in the bin and the passive state is taken in the hopper. The stress in the bin is seen to vary linearly while the stress in the hopper is linear at first, close to the outlet and then increases exponentially. This behavior is because the bin is relatively short and so the stresses do not reach their maximum, asymptotic value. One can also see the large stress peaks that occur at the joining point between the bin and the hopper. Overall, good agreement between theory and measurement can be observed.