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March 2021 / 31 tical conveying line). Another important velocity to be aware of is choking velocity, which is the gas velocity at which the particles conveyed upward in a verti- cal pipeline begin to approach their free-fall velocity. Approaching the particles' free-fall velocity results in the material no longer conveying upward but instead collapsing into a fluidized-bed (refluxing) condition. Commonly, an air velocity, u air , of 70 ft/s (4,200 fpm) is selected for many powders and granular solids in a dilute-phase conveying mode, so we'll use this as the minimum conveying velocity. However, selecting a suitable velocity is critical, since too low of a value can result in line plugging and too high of a value increases pressure drop at velocity squared (u 2 ) and the potential for line wear and particle attrition at velocity cubed (u 3 ). Ideally, pilot-scale testing with a representa- tive material subjected to an anticipated solids loading range will help to determine the appropriate minimum conveying velocity. For this example, assume the solids friction factor, λs, is 0.02 and the solids loading ratio, ϕ, is 7 (based on stable pneumatic conveying results). Assuming the conveying air is 70ºF at sea level condi- tions, use an air density, ρ air , of 0.075 lb m /ft 3 , where lb m is pounds of mass, and a dynamic air viscosity, μ air , of 0.000012 lb m /ft s. Step 1: Calculate air mass flowrate Using Equation 1, calculate the air mass flowrate, ṁ air , using the solids loading ratio, ϕ, 7, and the desired sol- ids mass flowrate, ṁ solids , of 10,000 lb/h. Ø = SLR = ṁ solids → ṁ air = ṁ solids ṁ air Ø 1 ṁ air = 10,000lb m /h = 1,429 lb m 7 h For a pneumatic conveying system to reliably operate in a stable manner, the solids feed into the conveying line must be controlled. 3 Frequently, prob- lems of nonuniform feed into the pneumatic conveying lines are thought to be problems with the pneumatic conveying system itself, whereas the problem actually lies with the upstream equipment. Feeding solids into a positive-pressure system requires a means of sealing against the pressure in the pipeline. Devices used for this purpose include rotary valves, double flap gates, specially designed sealing screws, and venturi nozzles. Some of these devices (rotary valves and sealing screws) control the solids flowrate into the line, hence they're truly feeders, while others (double flap gates and venturi nozzles) only provide a pressure seal but don't meter solids. Use the general solution algorithm steps below to estimate the system's pressure loss from conveying the air and solids. Ultimately, this calculation will help you to determine what air mover type is required to provide the necessary conveying velocity for the solids as well as the air mover's ability to respond to the pres- sure drop. 1. Calculate air mass flowrate. 2. Calculate pipeline diameter required for desired minimum conveying velocity. 3. Calculate total pressure drop. 4. Calculate new air density and velocity at pickup. 5. Recalculate air mass flowrate, pipeline diameter, and total pressure drop based on new air density. 6. Select a suitable air mover. Minimum conveying velocity Let's assume that through pneumatic conveying tests, the minimum conveying velocity, u min , for this material ranges from 60 to 70 ft/s over a range of solids loading conditions. This is perhaps the most important velocity for the pneumatic conveying system designer. This is the minimum velocity that must exist in a given pneu- matic conveying system for a given material to move to prevent plugging the pipeline. In dilute-phase con- veying, whereby all of the solids are suspended in the moving airstream, an air velocity of approximately 50 ft/s (3,000 fpm) is a lower bound for fine and low- density powders. Lower bound is the value that's less than or equal to every element of a data set. It's well-doc- umented that larger particles with higher particle densities require a higher minimum conveying velocity 2 to prevent saltation (when solids settle in a horizontal conveying line) and choking (when solids stall in a ver- FIGURE 1 An example of a positive-pressure dilute-phase pneumatic conveying system layout. Silo Filter-receiver 50 feet horizontally 30 feet vertically 50 feet horizontally