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16 October 2017 Tablets & Capsules where the shear stress at the wall τ' is calculated from Equation 8. Minimum outlet dimension To prevent the formation of a stable cohesive arch at the outlet of a hopper, the external stress must be greater than the powder's unconfined yield strength. Jenike [1] defined the flow factor ff as the ratio of the major consol- idation stress σ 1 to the stress on the abutment of the arch that naturally forms at the outlet σ: (14) The flow factor depends on the powder's effective angle of friction δ, the wall friction angle ϕ', and the hop- per angle θ'. Charts that provide flow factors are pro- vided by Jenike [1], and an example of a chart is given in Figure 7, which shows the flow factor for powders having an effective angle of friction equal to 50 degrees that are handled in conical hoppers. Analytical expressions of the flow factor are provided by Arnold and McLean [3,4]. For conical hoppers, the flow factor can be determined from: (15) where (16) (17) and (18) Y= [2(1-cos(β+θ'))]sinθ'+sinβsin2(β+θ') (1-sinδ)sin 3 (β+θ') σ 1 σ ff= ‾ ff= Y(1+sinδ)H(θ') 2(X-1)(sinθ') ] [ X= 2sinδ sin(2β+θ') sinα 1-sinδ H(θ')= 130˚+θ' 65 These equations may be serpentine, but they can read- ily be input into spreadsheets and other software. Flow factors typically range between 1.2 and 1.6 but can be much greater if wall friction is exceptionally low and hopper walls are very steep. The cohesive strength and arch stress can be com- pared by superimposing the flow factor and flow function on the same graph. The flow factor is constructed by drawing a line having a slope equal to 1/ff through the origin. As shown in Figure 8, three possibilities exist: 1. The flow function lies below the flow factor, and the two do not intersect. In this case, the stress imparted on the abutment of the arch is always greater than the material's cohesive strength, and there is no minimum outlet dimension requirement to prevent cohesive arch- ing. Instead, the outlet dimension B is determined by other considerations, such as the discharge rate required. The hopper angle required for mass flow requires the major consolidation stress σ 1 at the outlet to be known. This stress is calculated from: (19) σ 1 =ff ρ b gB H(θ') Figure 8 Plot of flow function and flow factor Unconfirmed yield strength or arch stress Major consolidation stress Flow function lies above flow function Flow factor slope = 1/ff Flow function intersects flow function Flow function lies below flow function Figure 9 Flowchart for determining critical hopper outlet size and mass-flow hopper angle Guess ff Calculate δ Calculate ϕ Determine σ 1 from intersection of flow factor and flow function Select θ' that allows mass flow (include 3˚ safety factor) Calculate H (θ')' Calculate new ff using updated values of δ ϕ' and θ' Calculate σ criti ρ b H (θ') Calculate B min ff converged? No ‾