Pressures in hoppers

The Janssen theory describes pressures in a parallel-sided vessel. The corresponding theory for a converging channel came much later, and is normally attributed to Walker (1964, 1966), though it was first derived by Dabrowski (1957) and was probably also found by Jenike and others in the late 1950s. 

Cancelling, eliminating small terms and noting that k2/k ) = l for both geometries 

The two assumptions used in the Janssen analysis are next made  

It is evident that the value of F must depend on geometry and solids properties, just as K was dependent on solids properties in the analysis of the pressures on vertical walls. The normal pressures may be deduced from Equation (3.29) as 

The question of whether the friction is fully mobilised in a hopper depends on its slope and the smoothness of the wall. The hopper is classed as steep if the solids slide on it, and this is met by the following test. The hopper is steep if 

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Commercial sensors are used to measure pressure, strain, acceleration, and sound propagation. These sensors are usually mounted on walls of granular experiments, for example, on hoppers [142] and shear cells [183]. However, there are sensors developed in the late 1950s, which can be placed into a granular material and report measurements wirelessly. The "rf-pill" was developed for medical applications [353-357] to measure the pressure in the digestive tract. The rf-pill is a small transmitter and sensor, which has been used to measure stress in vertical tube flow [158] and pressure in hopper discharge [128,133]. 

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