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2. The flow function lies above the flow factor and the curves do not intersect. The powder will not flow due to gravity alone. Consideration should be given to changing the flow properties of the material, perhaps by increasing its particle size or using a flow aid. 3. The flow function and flow factor intersect. At the intersection of the two lines, the arch stress and the cohe- sive strength of the bulk solid are the same and equal to the critical stress σ crit . The hopper outlet diameter that must be exceeded to prevent arching, B min can be calculated from: (20) The bulk density ρ b the effective angle of friction δ, and the angle of wall friction ϕ' depend on stress, and therefore calculating critical hopper angles and arching dimensions is an iterative procedure. Figure 9 is a flowchart for calcu- lating the minimum hopper outlet diameter and recom- mended mass-flow hopper angle for the case where the flow function and flow factor intersect. A flowchart for determining the recommended mass-flow hopper angle for selected outlet diameters is given in Figure 10. The outlet must be large enough to provide the desired discharge rate. For coarse powders, the solids dis- charge rate m s is given by: B min = H(θ')σ crit ρ b g (21) The maximum discharge rate of a fine powder can be orders of magnitude less than that of a coarse powder due to an adverse gradient that naturally develops near the outlet as the powder dilates. For fine powders, the follow- ing equation provides a conservative estimate of the max- imum discharge rate [5]: (22) where the subscripts o and max denote the value at the outlet and the maximum value, respectively. The maxi- mum bulk density ρ bmax develops at the major consolida- tion stress at the hopper cylinder junction, which is cal- culated from the Janssen equation [6]: (23) where D is the cylinder diameter, h is the depth of pow- der in the cylinder section, and k is the Janssen coeffi- cient. Average values of the bulk density and wall friction angle are used. The Janssen coefficient can be assumed equal to 0.4. Sample calculations Consider the powder blend containing 67 percent APAP, 29 pecent MCC, and 4 percent HPC. Its powder flow properties appear in figures 3 and 4, and follow these steps to design a suitable mass-flow hopper: 1. Choose 1.2 as an initial estimate for ff. 2. Determine the major consolidation stress at the inter- section of the flow function and flow factor. The major consolidation stress σ 1 is equal to 1.07 kilopascals (kPa). ) ( 1- m s = ρ bo ρ bo ρ bmax 4 πB 2 K 0 ̇ ) ( 1-exp ] [ σ 1 = ρ b gD _ 4ktan(ϕ')h D 4ktan(ϕ') m s = ρ bo 4 √4tan(θ') Bg πB 2 ̇ Tablets & Capsules October 2017 17 Figure 11 Flow function of APAP-MCC-HPC blend with 0.2 percent fumed silica and 1.0 percent magnesium stearate Major consolidation stress (kPa) Unconfirmed yield strength (kPa) 0 2 4 6 8 10 5 4 3 2 1 0 Figure 10 Flowchart for determining recommended mass- flow hopper angle for a specified outlet diameter Choose B Guess δ, ϕ' Calculate ff Select θ' that allows mass flow (include 3˚ safety factor) Solve σ 1 = Calculate H (θ') θ' is recommended value Update δ, ϕ' δ, ϕ' converged? No ffρ b gB H (θ') ˙