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angle of friction angle of internal assume bulk solid chute compacting pressure conical surface considered cosh 2k coth determined Differentiating Eliminating ellipse equation factor fc/w Figure 15 Figure 17 Figure 20 Figure 9 flow of bulk flow-factor curve flowability formula friction between material funnel flow gate or feeder hoop pressure hopper walls horizontal hyperbolas impact incipient failure increases influence instability largest mass flow mass of material material and wall material flows minor axis model bins obtain plane of cleavage pres pressure q pressure v/w principal pressures ratio L/B result shearing stress shown in Figure sinh 2k slope angle SOLIDS IN BINS spout stable dome static pressure steep hoppers strength of material surface tanh terial tests unconfined compressive strength unit pressure unit strength unit weight University of Utah vertical portion vertical pressure vertical walls vibration wedging action width zero