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ϕ', and the geometry of the hopper. Figure 7 provides mass flow boundaries for conical hoppers based on analy- ses developed by Jenike [1]. Values of the allowable hop- per angle θ' are on the horizontal axis, and values of the angle of wall friction ϕ' are on the vertical axis. The theo- retical boundaries between mass flow and funnel flow depend on the effective angle of friction δ. Any combina- tion of θ' and ϕ' that falls within the limiting mass flow region of the chart will provide mass flow. A 2- to 3- degree safety factor with respect to the theoretical mass- flow hopper angle is recommended when designing a conical mass-flow hopper. The figure confirms that mass flow is more likely in hoppers with steep walls (small val- ues of θ') and low friction (small values of ϕ'). An analytical expression for the recommended mass flow boundary is given by Equation 2 [2]: (2) where (3) Note that the equation includes a 2- to 3-degree mar- gin of safety. The wall friction angle ϕ' depends on the effective angle of friction δ and the major consolidation stress σ 1 . It is determined from the intersection of the Mohr's circle associated with σ 1 and δ and the wall yield locus. See Figure 6. The Mohr's circle is given by (4) where (σ-σ avg ) 2 +τ 2 =R 2 θ'=90º- -β-(2-3º) 1 2 cos -1 1-sinδ 2sinδ ) ( ϕ'+sin -1 β= 1 2 sinϕ' sinδ ) ) ( ( (5) and (6) The minor principle stress σ 2 is related to the effective angle of friction δ by: (7) If the wall yield locus is linear, which is often true at low stresses, it can be described by: (8) where τ' and σ' are the shear and normal stresses at the wall surface, respectively, and a and b are empirical con- stants determined from regression. The normal stress can then be calculated from: (9) where (10) (11) and (12) The wall friction angle is then calculated from: (13) -β+√β 2 -4αγ 2α σ'= τ'=aσ'+b 1-sin(δ) 1+sin(δ) σ 2 = σ 1 σ avg = σ 1 +σ 2 2 R= σ 1 -σ 2 2 α=a 2 +1 β=2(ab-σ avg ) γ=b 2 +σ 2 avg -R 2 τ ' σ ' ϕ'=tan -1 ) ( 14 October 2017 Tablets & Capsules Figure 6 Determination of the wall friction angle from the major consolidation stress and effective angle of friction Shear stress τ' 0 σ 2 σ' σ 1 Normal stress Effective yield locus Wall yield locus Figure 7 Flow factor chart for conical hoppers, = 50 degrees Wall friction angle (deg) 0 10 20 30 40 50 50 40 30 20 10 0 Hopper angle from vertical (deg) ff = 1.6 ff = 2.0 ff = 3.0 ff= 4.0 ff= 1.4 ff= 1.3 ff= 1.2 δ= 50˚