Stresses in Bins and Hoppers, by Gabriel I. Tardos

-- 7. Calculating the Flow Factor --

The expressions in Eq. 28 and in the Appendix can be used directly to calculate the flow factor ff. These are however only some of many equations and charts dedicated to the prediction of this variable. The most detailed calculations for the flow factor were done by Jenike [1964] who solved numerically the differential equations of equilibrium of a bulk material in the hopper and gave the results in the now famous charts that bear his name. Analytical expressions such as the one given in Eq. 28 are used only in as far as they agree with the more precise values calculated by Jenike. An example of a Jenike chart is given in Fig. 12 for a plane and a conical hopper for a material with an internal angle of friction of = 40o.

Fig. 12.a - Values of the flow factor for a 2D plane hopper.
Material angle if internal friction angle = 40^o

Fig. 12.b - Values of the flow factor for a conical hopper.
Material angle if internal friction angle = 40^o

Similar charts for other angles of friction from 30o - 70o and various geometries can be found in the above-mentioned work. There is a slight difference in the design Eq. 30 when using Jenike's charts in that the geometric constant, m, is replaced by a more complex function, H() given in Fig. 13. Fig. 13 - Function H() for 2D and 3D hoppers As seen, H()m for very small values of the hopper half-angle, , but deviates from it at larger values.