Issue link: https://www.e-digitaleditions.com/i/1344205
32 / March 2021 powderbulk.com where ΔP air is the pressure loss from the air moving through the pipeline, ΔP solids is the pressure loss from the solids acceleration, and ΔP equipment is the pressure loss from one piece of equipment to another (for exam- ple, the piping from the blower to the feedpoint or a diverter valve in the conveying line). Again, because that pressure loss is such a small amount compared to the rest of the pressure losses and this is a simpli- fied equation, we'll disregard it for this equation. The pressure loss for the air moving through the pipeline can be found using the Darcy-Weisbach relationship shown in Equation 4. The "knowns" in this equation are velocity, u air ; air density, ρ air ; and pipeline internal diameter, D, from Equation 2. The "unknowns" are air friction factor, ƒ, and equivalent line length, L. ΔP air = f Lρ air u air 2 2D 4 The air friction factor, ƒ, can be determined through the use of the Moody diagram 4 or the Colebrook equa- tion for laminar and turbulent flow through pipes, where the relative pipe internal roughness and Rey- nold's number are known. The relative roughness is the ratio of absolute roughness to pipeline diameter. A typical value for absolute roughness for a smooth bore steel pipeline is 0.0005 feet. Thus, a relative roughness of 0.0017 feet results from this example. The Reynold's number, Re, which is the non-dimensional ratio of air inertial-to-viscous forces, can be calculated with Equation 5. Re = u air ρ air D μ air 5 Re = = 1.44 x 10 5 77 ft 0.075 lb m 3.55 in s ft 3 12 in ⁄ft 1.2 x 10 −5 lb m ft ⋅ s Remember, D now equals the pipeline's internal diameter of 3.548 inches, which has been rounded up to 3.55 for this article. Using the Reynold's number and the relative roughness, a pipeline air friction factor of 0.023 is estimated. The length term in Darcy's equation (4) can be either equivalent or actual pipeline length, depending on the calculation need. The total equivalent length, Leq, accounts for horizontal and vertical pipelines that the air will flow through, plus an estimated equivalent length for short and long radius bends. For this exam- ple with two elbows having a radius-to-diameter ratio Step 2: Calculate pipeline diameter Using Equation 2, which is the law of continuity from fluid mechanics, calculate the pipeline diameter, D, with the specified minimum conveying velocity, u min , of 70 ft/s. The law of continuity means that what flows into a defined volume in a defined time, minus what flows out of that volume in that time, must accumulate in that volume. ṁ air = ρ air A pipe u air = ρ air ( π D 2 ) u min 4 2 D = 4ṁ air √ ρ air u air π (4) 1,429 lb m h 0.075 lb m 70 ft 3,600 s π √ ft 3 s h D D = 0.31ft → 3.72in. (94.5mm) where A pipe is the pipeline cross-sectional area. When applying the 3.72-inch pipe diameter results to ANSI's steel pipe specification schedule 40, we see that we need to select a 3.5-inch schedule 40 pipe with an inter- nal diameter equal to 3.548 inches (rounded up to 3.55 inches for simplicity). Proceeding with the calculations, D now represents internal diameter, which is 3.55 inches. At this smaller internal diameter, the air veloc- ity, u air , will be slightly higher (77 ft/s) based on the ratio of the diameters squared. Step 3: Calculate total pressure drop The major forces involved in dilute-phase conveying of bulk solids in an airstream are: • Friction between moving air against pipeline wall • Friction between moving solids against pipeline wall • Force required to move air through vertical pipe- line sections • Force required to lift solids through vertical pipe- line sections • Force required to accelerate solids from feedpoint velocity to conveying velocity In consideration of these forces, Equation 3 provides the total pressure drop, ΔP total , for a system, including a term for additional losses, such as with an air and solids separator. For this example, we'll assume the pressure loss for the equipment is negligible. ΔP total = ΔP air + ΔP solids + ΔP equipment 3