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June 2021 / 15 ṁ s =ρ bo A o Bg 1+ 1 dP √ 2(m+1)tanϴ′ ρ bo g dz o 12 where m is equal to 1 for conical hoppers and equal to 0 for hop- pers with straight walls and slotted outlets. Because the gas pressure gra- dient at the outlet, dP/dz| o , is less than zero, Equation 12 shows that fine powders can have dramatically lower discharge rates than coarse powders. A fine powder's maxi- mum flowrate can be several orders of magnitude lower than that of coarser materials. Two-phase flow effects are significant due to the movement of interstitial gas as the powder compresses and expands during flow. Solids and gas pres- sure profiles in bins for coarse (high permeability) and fine (low permeability) powders are shown in Figure 3. In a bin's straight-walled sec- tion, the stress level increases with depth, causing the material's bulk density to increase and its void fraction to decrease, squeezing out a portion of the interstitial gas. This gas leaves the bulk material through its top, free surface. To flow in a bin's hopper section, the consolidated material expands as it moves toward the outlet, reduc- ing its bulk density and increasing its void fraction. This expansion results in a reduction of the inter- stitial gas pressure to below atmospheric (that is, a vacuum), causing gas counterflow through the outlet if the pressure below the outlet is atmospheric. A vacuum will develop even if the bin's top is vented. At a critical solids discharge rate, the solids contact pressure reduces to zero, and efforts to exceed this limiting discharge rate will result in erratic flow. The pressure gradient is related to the material's permeability and the gas slip velocity, u g , by Darcy's law If the bulk density is assumed constant d Av = 0 dz 4 where A is the cross-sectional area of the hopper outlet. Therefore v 2 dA = g + 1 dP A dz ρ b g dz 5 For round outlets A = πr 2 = π (z tan ϴ′) 2 6 dA = 2π ztanϴ′ dz 7 At the hopper outlet 1 dA = 4tanϴ′ A dz B 8 where B is the outlet diameter and ϴ' is the hopper angle (from verti- cal). Therefore 4v 2 o tanϴ′ = g + 1 dP B ρ b g dz 9 and v o = Bg 1 + 1 dP √ 4tanϴ′ ρ bo g dz o 10 where the subscript o denotes the hopper outlet. A similar analysis for planar hoppers with flat walls and slotted outlets yields v o = Bg 1 + 1 dP √ 2tanϴ′ ρ bo g dz o 11 where B is the outlet width. The discharge rate, ṁ s , is the product of the powder's velocity, its bulk density at the outlet, ρ bo , and the cross-sectional area of the out- let, A o . Thus coarse powder's steady-state dis- charge rate from a hopper can be determined from a force balance. Consider a hopper with the geometry shown in Figure 2. An equilibrium force balance on the powder inside the hopper is given by ρ b a = − ρ b g − dP dz 1 where a is the powder's accelera- tion, ρ b is its bulk density, g is the acceleration due to gravity, P is the interstitial gas pressure, z is the axial coordinate (the origin is the hopper's apex), and dP/dz is the gas pressure gradient. To be able to cal- culate a solids discharge rate, we'll need to derive an equation that pro- vides a velocity, so we employ the following trick a = dv = dz dv = v dv dt dt dz dz 2 where v is the powder's velocity, and t denotes time. Hence, Equation 2 can be rewritten as v dv = − g − 1 dP dz ρ b g dz 3 FIGURE 2 Hopper geometry B z ϴ'