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March 2021 / 35 P new is the new pressure calculated from P 2 / P 1 ; and P atm is atmospheric pressure. ρ 1 A 1 u 1 = ρ 2 A 2 u 2 13 u new = u air ρ air A 1 = 77 ft 0.075 lb m /ft 3 = 59.6 ft/s ρ air,new A 2 sec 0.097 lb m /ft 3 where A 1 and A 2 are the pipeline cross-sectional area at the system's start and at the solids feedpoint, respec- tively; and u 1 and u 2 are the air velocity at system start and the air velocity at the solids feedpoint, respectively. Note that the recalculated velocity, u new , is lower than the initial specified velocity of 70 ft/s. This demon- strates the compressibility effects for the air, which can also be affected by temperature changes as well. If testing has shown that 70 ft/s is required to avoid line buildup and plugging, then you'll be required to revisit this algorithm with a higher initial velocity to compen- sate for the air compression (and reduced volume and velocity at the pickup point). The pickup velocity is the gas velocity at the pickup point of the conveying sys- tem where solids are introduced into the conveying gas stream. When used in a research connotation, this is the gas velocity necessary to pick up, suspend, entrain, or detach particles at rest in a settled layer on the bot- tom of a horizontal pipeline. If the air velocity is calculated to be too high (espe- cially toward the end of the line due to the air density dropping), consider line stepping. Line stepping can be achieved by increasing the pipeline's internal diam- eter at discrete points, thereby locally reducing the air velocity. Be cautious to not reduce the velocity to a point where either saltation occurs or an unstable con- veying regime exists. Step 5: Recalculate air mass flowrate, pipeline diameter, and total pressure drop based on new air density A more accurate calculation method involves an itera- tive procedure that's performed in discrete steps along the pipeline to account for the change in air density and velocity. For brevity, the reiteration isn't covered in this article. Step 6: Select a suitable air mover Now that we have determined the rates for each vari- able, we can figure out the volumetric airflow rate. The volumetric airflow rate, Q, can be found with Equation 14, assuming approximately 60 ft/s velocity is sufficient. We can assume this because as stated at the beginning of this article, the minimum conveying velocity for this material ranges from 60 to 70 ft/s. 1,014 lb m ΔP lift = ft⋅s 2 = 31.5 lb f ft 2 = 0.22 lb f = 0.22 psi 32.2 lb m ⋅ft ft 2 12 in in 2 lb f ⋅s 2 11 Now the solids pressure loss can be totaled, as below. ΔP solids = ΔP accel. + ΔP friction + ΔP bends + ΔP lift 7 ΔP solids = 0.34 psi + 2.95 psi + 0.20 psi + 0.22 psi = 3.71 psi Then, the total pressure loss, ΔP total , can be found from the solids and air components together. As stated previously, assume the pressure loss for the equipment is negligible. ΔP total = ΔP air + ΔP solids + ΔP equipment 3 ΔP total = 0.52 psi + 3.71 psi = 4.23 psi Additional pressure loss could be included due to the air and solids separator, additional piping before the solids infeed, or after solids disengagement. Typi- cally, for air and solids separators, a pressure loss of 6 inches water gauge is considered. Step 4: Calculate new air density and velocity at pickup After the total pressure drop is calculated, the air velocity, u air , at the solids' feedpoint needs to be recal- culated due to the higher pressure than initially assumed at atmospheric conditions and corresponding air density. Equations 12 and 13 provide the new air density, ρ air,new , and new air velocity, u new , and calcula- tions are based on the pressure drop determined in step 3. Remember to use absolute pressure and tem- perature (when appropriate) in these calculations. ρ = P → ρ 2 = P 2 /RT 2 RT ρ 1 P 1 /RT 1 →ρ 2 = ρ 1 P 2 P 1 12 ρ air,new = ρ air P new = 0.075 lb m 14.7 psi + 4.25 psi = 0.097 lb m P atm ft 3 14.7 psi + 0 psi ft 3 where ρ 1 is the initial air density at the system's start and ρ 2 is the air density recalculated at the solids feed- point; P 1 is the initial pressure at the system's start and P 2 is the pressure recalculated at the solids feedpoint; RT is the gas constant (R) and the gas temperature (T), and RT 1 and RT 2 refer to locations in control volume;