particulate-collection

name	particulate-collection
description	Particulate collection — cyclone separator (Lapple model, cut diameter d50), electrostatic precipitator (Deutsch-Anderson equation, SCA, migration velocity), baghouse filtration (Kozeny-Carman, pressure drop vs. areal density), scrubber (Venturi, contacting power), gravity settler (terminal velocity, Stokes law), particle size distribution (log-normal, d50/d84), collection efficiency grade curve, EPA 40 CFR Part 60 Method 5, and ACGIH industrial ventilation standards.
metadata	{"priority":7,"promptSignals":{"phrases":["cyclone separator","electrostatic precipitator","baghouse","particulate collector","dust collection","particle separation"],"minScore":3}}

Particulate Collection — Complete Skill

Particle Fundamentals

Particle Size and Aerodynamic Diameter

Stokes law (creeping flow; Re_p < 1): F_drag = 3π × μ × V_s × d_p × C_c [N; V_s = slip velocity; d_p = particle diameter; C_c = Cunningham correction]

Cunningham slip correction (for d_p < 10 μm): C_c = 1 + (2λ/d_p) × [1.257 + 0.400 × exp(-0.55 × d_p/λ)] λ = mean free path of gas ≈ 0.066 μm (air at 20°C, 1 atm) Significant for d_p < 1 μm; C_c ≈ 1 for d_p > 5 μm

Terminal settling velocity (gravity): V_t = ρ_p × d_p² × g × C_c / (18 × μ) [m/s; ρ_p = particle density; Stokes regime Re_p < 1] V_t = d_p² × ρ_p × g / (18 × μ) × C_c [simplified for Re_p << 1]

Newton's regime (Re_p > 1000): V_t = √(4×d_p×ρ_p×g / (3×C_D×ρ_g)); C_D = 0.44 Intermediate (1 < Re_p < 1000): C_D = 24/Re_p + 6/(1+√Re_p) + 0.4 [Schiller-Naumann]

Aerodynamic diameter d_ae: d_ae = d_p × √(ρ_p × C_c / (ρ₀ × C_c,ae)) [ρ₀ = 1,000 kg/m³ reference; equivalent Stokes diameter of unit density sphere]

Log-normal particle size distribution: f(d_p) = 1/(d_p × σ_g × √(2π)) × exp(-[ln(d_p/d_50)]² / (2×σ_g²)) d_50 = mass median diameter (MMD); σ_g = geometric standard deviation (GSD) For σ_g = 2: d₁₆ = d_50/2; d₈₄ = 2×d_50 (68% within factor of 2)

Gravity Settlers

Settling Chamber Design

Collection efficiency (plug flow, gravity settler): η = V_t × A_floor / Q_gas [A_floor = settler floor area; Q_gas = volumetric gas flow] η = 1 for V_t × L/(H × u_g) ≥ 1 [L = length; H = height; u_g = gas velocity]

Cut diameter (η = 50%): d_50 = √(9 × μ × H × u_g / (ρ_p × g × L × C_c)) [rearranged from η formula]

Design guidelines: u_g < 3 m/s (to avoid re-entrainment); L/H ≥ 5 typical Effective only for d_p > 50–100 μm; impractical for fine dust

Cyclone Separators

Lapple Model

Number of effective turns (standard cyclone): N_e = (H_b + H_c/2) / (2 × B) [H_b = barrel height; H_c = cone height; B = barrel width; typically N_e = 5]

Cut diameter (d_pc at 50% efficiency): d_pc = √(9 × μ × B / (π × N_e × V_i × ρ_p)) [m; V_i = inlet gas velocity; B = body width]

Fractional efficiency (Lapple): η(d_p) = 1 / [1 + (d_pc/d_p)²] [S-curve; η = 50% at d_p = d_pc; η > 90% for d_p > 2×d_pc]

Pressure drop: ΔP = N_H × ρ_g × V_i² / 2 [N_H = dimensionless loss factor; N_H = 8 for standard cyclone; 4 for high-efficiency] ΔP ≈ 750–1,500 Pa typical for industrial cyclone at V_i = 15 m/s

Standard cyclone proportions (Lapple standard): D = body diameter; H_i/D = 0.5; B_i/D = 0.25; D_e/D = 0.5; L_e/D = 0.625; H_b/D = 1.5; H_c/D = 2.5; D_u/D = 0.25

Example: Gas: Q = 1 m³/s, μ = 1.8×10⁻⁵ Pa·s, ρ_g = 1.2 kg/m³ Particle: ρ_p = 2,000 kg/m³; B = 0.5 m; N_e = 5; V_i = 15 m/s d_pc = √(9×1.8×10⁻⁵×0.5/(π×5×15×2000)) = √(8.1×10⁻⁵/471,239) = √(1.72×10⁻¹⁰) = 13.1 μm

High-efficiency cyclone: smaller D (0.25–0.5 m); higher V_i; d_pc = 2–5 μm; multiple units in parallel

Electrostatic Precipitators (ESP)

Deutsch-Anderson Equation

Collection efficiency: η = 1 - exp(-w_e × A_total / Q) [Deutsch-Anderson; η = collection efficiency] w_e = migration velocity (drift velocity) [m/s]; A_total = total collection plate area [m²]; Q = gas flow [m³/s]

Specific collection area (SCA): SCA = A_total / Q [m²/(m³/s) = s/m] Typical: SCA = 60–120 s/m for utility fly ash (η = 99–99.9%)

Migration velocity w_e: w_e = q_p × E_c / (3π × μ × d_p) [Stokes-Millikan; q_p = particle charge; E_c = collecting field strength] Typical w_e = 0.05–0.30 m/s for industrial fly ash; function of resistivity

Particle charging: Field charging (d_p > 1 μm): q = π × ε₀ × (1 + 2(ε_p-1)/(ε_p+2)) × d_p² × E_c Diffusion charging (d_p < 0.2 μm): q = (d_p × k_B × T)/(2e) × ln(1 + π×e²×N_i×c̄×t/(2×k_B×T)) [N_i = ion number density; c̄ = mean thermal speed of ions; t = charging time]

Resistivity effects: Low resistivity (< 10⁸ Ω·cm): poor charge retention → re-entrainment High resistivity (> 10¹¹ Ω·cm): back-corona; breakdown; reduces collection Optimal: 10⁸–10¹⁰ Ω·cm Conditioning: SO₃ injection (1–15 ppm) reduces fly ash resistivity

SCA required for target efficiency: SCA = -ln(1-η) / w_e [from Deutsch-Anderson rearranged] For η = 99.5%, w_e = 0.10 m/s: SCA = -ln(0.005)/0.10 = 53 s/m

Baghouse (Fabric Filters)

Filtration Mechanism and Pressure Drop

Kozeny-Carman model (cake filtration): ΔP = μ × w × α × u_f + μ × S_e × u_f [Pa; w = areal density of dust cake [kg/m²]; α = specific cake resistance; u_f = face velocity [m/s]] S_e = fabric resistance [1/m]; typically S_e = 5×10⁸ – 5×10⁹ m⁻¹

Specific cake resistance α: α depends on particle size, shape, compressibility: typically 10¹⁰–10¹² m/kg

Air-to-cloth (A/C) ratio: u_f = Q / A_fabric [m/min or m/s] Pulse-jet: u_f = 2–4 m/min (0.033–0.067 m/s); reverse-air: 0.8–1.2 m/min; shaker: 0.7–1.2 m/min

Filtration efficiency: Fabric filters achieve η > 99.9% for d_p > 1 μm; η > 99% for PM2.5 with proper design Mechanisms: interception, inertial impaction, diffusion (dominant for d_p < 0.3 μm), electrostatic

Pulse-jet cleaning cycle: Interval between pulses: t_clean = (ΔP_max - ΔP_min) / (dΔP/dt) [s] dΔP/dt = μ × α × ρ_dust × u_f × C_inlet [Pa/s; C_inlet = inlet dust concentration kg/m³]

Typical operating parameters: Inlet loading: 1–100 g/m³; outlet: < 10 mg/m³ (EPA limits); pulse pressure: 400–700 kPa

Venturi Scrubbers

Contacting Power Theory (Lapple-Shepherd)

Efficiency vs. contacting power: η = 1 - exp(-N × P_contacting^β) [N, β = empirical constants by dust type] P_contacting = ΔP_gas + L × ΔP_liquid/Q_gas [kPa; L/Q = liquid-to-gas ratio L/m³]

Venturi pressure drop: ΔP_gas = ρ_g × V_throat² × (1 + L/G × ρ_L/ρ_g)/2 [V_throat = throat gas velocity; L/G = liquid-gas ratio] Typical V_throat = 60–120 m/s; ΔP = 1–15 kPa

d_50 for Venturi (Calvert model): d_50 = √(55 × μ_g × D_throat / (ρ_p × V_rel × Q_L/Q_g × 100)) V_rel = V_throat (droplet velocity ≈ 0 at contact)

Liquid-to-gas ratio: L/Q_g = 0.5–2.0 L/m³ typical for medium-efficiency scrubbers High energy (ΔP > 10 kPa): can achieve d_50 < 1 μm

Grade Efficiency Curves

Overall Collection Efficiency

Weighted efficiency: η_overall = Σ η(d_pi) × Δw_i [Δw_i = mass fraction in size range centered at d_pi] Computed from PSD and grade efficiency curve

Grade efficiency from log-normal PSD: η_overall = ∫ η(d_p) × f(d_p) × dd_p [numerical integration]

Particle sampling — EPA Method 5: Isokinetic sampling: V_nozzle = V_stack (velocity matched → representative sample) Method 5: heated filter + impingers at 120°C → catch PM total Method 17: in-stack filter; PM at stack temperature Method 201A: cyclone + filter for PM10/PM2.5 classification

Standards and References

Standard	Scope
EPA 40 CFR Part 60 (NSPS)	New source performance standards; PM emission limits
EPA Method 5	Stack gas particulate sampling
ACGIH Industrial Ventilation	Design manual for industrial exhaust systems
ASHRAE 52.2	MERV rating for HVAC filters
ISO 16890	Fine dust filtration test for air filters (ePM1/ePM2.5/ePM10)
ASME PTC 28	Determining dust concentration in gas streams

Output

Provide: particle characterization (d_50 [μm]; GSD σ_g; ρ_p [kg/m³]; resistivity [Ω·cm] for ESP), gas stream (Q [m³/s]; T [°C]; μ [Pa·s]; inlet concentration C [g/m³]), selected collector type (cyclone/ESP/baghouse/scrubber) with justification, cyclone — cut diameter d_pc [μm]; grade efficiency curve; ΔP [Pa]; N_e; body diameter D [m], ESP — Deutsch-Anderson: w_e [m/s]; SCA [s/m]; A_plates [m²]; required transformer-rectifier power [kVA], baghouse — A/C ratio [m/min]; ΔP_clean/dirty [Pa]; filter area [m²]; cleaning interval [min]; bag material, scrubber — throat velocity [m/s]; L/G ratio [L/m³]; ΔP [kPa]; d_50 [μm] from Calvert, overall collection efficiency η [%] (from grade curve × PSD), outlet loading [mg/m³] vs. regulatory limit, and applicable standard (EPA Method 5, ACGIH Industrial Ventilation, ASME PTC 28).