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Tablets & Capsules October 2017 13 low velocity. The superficial gas velocity is related to the gas pressure gradient by: (1) where u s is the gas superficial velocity; K is the powder's permeability; ρ b is its bulk density; g is equal to accelera- tion due to gravity; and dP/dz is the gas pressure gradient. Permeability test results are useful for estimating solids discharge rates. Many investigators use the metric FFC—the ratio of the major consolidation stress σ 1 to the unconfined yield strength f c determined from yield locus measurements—as a metric for flowability. FFC is frequently misused because it is generally evaluated at high stresses, where FFC is more readily distinguishable. But in reality, the stresses are low at the outlet of hoppers designed to prevent rathol- ing. In addition, the ratio ignores the influences of wall u s =- K dP ρ b g dz friction and bulk density on flow behavior. A better approach to optimize a formulation is to use the measured flow properties in order to determine the size of the outlet of a hopper that will prevent flow obstructions, the slope of its walls that will prevent ratholing, and the outlet size that will achieve the desired discharge rate. Mass-flow hopper angle Two flow patterns can occur in a hopper: mass flow and funnel flow. In mass flow, the entire bed of solids is in motion when material is discharged from the outlet. This behavior eliminates the formation of ratholes in the vessel, affords a "first-in, first-out" flow sequence, and provides a more uniform velocity profile during opera- tion. A uniform velocity profile mitigates sifting segrega- tion, which would result in side-to-side separation of par- ticles by size. In funnel flow, an active flow channel forms above the outlet, with stagnant material (i.e., ratholes) remaining at the periphery of the hopper. Hoppers in which funnel flow occurs may require a very large outlet to ensure that the ratholes collapse and the hopper empties. Funnel flow can cause erratic flow and exacerbate segregation, and material that forms the ratholes may spoil or cake. Mass- flow hoppers are therefore preferable for handling phar- maceutical formulations. The hopper angle required to allow mass flow depends on the effective angle of friction δ, the wall friction angle ̇ Figure 4 Wall friction of APAP-MCC-HPC blend on 304 stainless steel with 2B finish 5 4 3 2 1 0 Shear stress (kPa) 0 1 2 3 4 5 Normal stress (kPa) Figure 5 Theoretical mass flow boundaries. A safety factor of 3 degrees is recommended. 0 10 20 30 40 50 60 Hopper angle from vertical (deg) 50 45 40 35 30 25 20 15 10 5 0 Wall friction angle (deg) δ = 30˚ δ = 40˚ δ = 50˚ δ = 60˚