Air Pollution Control, Particulate Separation & Chemical Engineering
Lapple Cyclone Cut-Diameter (d50) & Grade Efficiency Formula
Calculate Lapple Reverse-Flow Cyclone 50% Cut-Size Diameter d_p50 (µm), Fractional Grade Collection Efficiency (%), Dust Penetration (%), and 8-Velocity-Head Pressure Drop (Pa).
Turning down boiler gas flow hurts cyclone dust capture—lower inlet velocity coarsens d_p50!
Unlike baghouses or electrostatic precipitators (which become MORE efficient when gas velocity drops), a centrifugal reverse-flow cyclone relies on vortex inertia: Lapple’s 50% Cut-Size Diameter is d_p50 = SQRT((9 * mu * B_c) / (2 * PI() * N_e * V_i * rho_p))! When a plant turns down airflow by 50%, inlet velocity V_i drops in half, coarsening d_p50 by SQRT(2) = +41.4% and allowing fine particulate to escape up the vortex finder according to Lapple’s Fractional Grade Efficiency eta_j = 1 / (1 + (d_p50 / d_p)^2)!
Verify on a Standard Lapple Cyclone—Gas_Viscosity mu = 0.000018 Pa·s (B2), Inlet_Width B_c = 0.25 m (B3), Effective_Vortex_Turns N_e = 5.0 (B4), Inlet_Velocity V_i = 16.0 m/s (B5), Net_Particle_Density = 2,500 kg/m³ (B6), and Target_Particle_Size d_p = 15.0 µm at Gas_Density = 1.20 kg/m³ and 8.0 velocity heads (B7): Cut-Size d_p50 equals 5.68 µm (E2), 15 µm Grade Collection Efficiency hits 87.47% (E3, with 12.53% penetration E4), and Cyclone Pressure Drop is 0.5 * 1.20 * 16^2 * 8 = 1,228.80 Pa (E5)!
| Lapple Cyclone Metric (Bc = 0.25 m, Ne = 5, Vi = 16 m/s, rho_p = 2,500 kg/m³) | Formula | Verified Output |
|---|---|---|
| 50% Cut-Size Diameter d_p50 (µm) | 10^6 * SQRT(9*mu*Bc / (2*PI()*Ne*Vi*rho_p)) | 5.68 µm (8.03 µm @ 8 m/s) |
| 15 µm Grade Efficiency (%) | (1 / (1 + (d_p50 / d_p)^2)) * 100 | 87.47% (77.73% @ 8 m/s) |
| 15 µm Particulate Penetration (%) | 100 - Grade_Efficiency_Pct | 12.53% (22.27% @ 8 m/s) |
| 8-Head Pressure Drop (Pa) | 0.5 * 1.2 * Vi^2 * 8 | 1,228.80 Pa (307.20 Pa @ 8 m/s) |
Lapple Cut-Diameter & Fractional Grade Efficiency Formula
Validate Viscosity > 0, Inlet_Width > 0, Turns >= 1, Inlet_Velocity > 0, Particle_Density >= 100, and Target_Diameter > 0!
=LET(mu_g, B2, bc_w, B3, ne_t, B4, vi_s, B5, rh_p, B6, dp_u, B7, IF(OR(mu_g <= 0, bc_w <= 0, ne_t < 1, vi_s <= 0, rh_p < 100, dp_u <= 0), "Invalid Cyclone Input", LET(d_50, 1000000 * SQRT((9 * mu_g * bc_w) / (2 * PI() * ne_t * vi_s * rh_p)), ROUND((1 / (1 + (d_50 / dp_u) ^ 2)) * 100, 2))))50% Turndown Velocity Loss Check (Set Inlet Velocity V_i = 8.0 m/s)
Halve Inlet_Velocity B5 from 16.0 m/s to 8.0 m/s: Cut-Size d_p50 E2 coarsens from 5.68 µm to 8.03 µm, and 15 µm Collection Efficiency E3 falls from 87.47% to 77.73% (nearly doubling dust emissions from 12.53% to 22.27% penetration)!
=IF(ISTEXT(E3), E3, ROUND(0.5 * 1.2 * (B5 ^ 2) * 8, 2))Try the example
Use the original values or change them to test your understanding.