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Radial profile: field E (MV/m, orange) and ion charge density (uC/m3 / 10, blue) against r (cm)
Particle charge (elementary charges) against time in the wall-region ion cloud, three sizes
Each button runs 8 to 11 particle simulations (260 particles each) and the Deutsch formula for comparison; it takes a few seconds.
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How to use
The gas flows upward with dust that starts grey. Press Guided tour for a narrated walk-through, pick a preset (below onset, high voltage, fine or coarse dust, fast gas, hot gas, back corona, ideal Deutsch model) or change any slider. Blue dots are ions from the corona, grains turn violet as they pick up charge, brown dots on the wall are collected dust. The readouts show voltage, current and the efficiency from a 500-particle run. Space pauses, R resets.
The science
What it does. An electrostatic precipitator removes dust from flue gas in three steps: (1) a high voltage on a thin wire makes a corona that fills the tube with ions; (2) the dust grains collect these ions and become charged; (3) the electric force drives the charged grains to the grounded wall, where they stick and are later knocked off into a hopper. Here the geometry is a wire of radius r_w on the axis of a tube of radius R.
Corona onset. The field of a bare wire is E = V / (r ln(R/r_w)). Corona starts when the field at the wire reaches Peek's empirical value, with relative gas density d = 293 / T and r_w in cm,
E_c = 3.1e6 d (1 + 0.0308 / sqrt(d r_w)) V/m, V_c = E_c r_w ln(R / r_w)
Space-charge field. Above onset the ions of density rho(r) drift outward at mu E; the current per unit length I = 2 pi r rho mu E is the same at every radius. With Gauss's law (1/r) d(rE)/dr = rho / eps0 this gives exactly
E(r)^2 = (E_c r_w / r)^2 + (I / (2 pi eps0 mu)) (1 - (r_w / r)^2), V = integral of E dr from r_w to R
The field at the wire stays clamped at E_c; the current I is found by solving for V numerically. The ion mobility is taken as mu = 2.0e-4 m2/(V s).
Field charging (Pauthenier). A particle of diameter d and permittivity eps_r in a field E collects ions until the saturation charge
q_s = 4 pi eps0 (d/2)^2 E * 3 eps_r / (eps_r + 2), dq/dt = (rho mu q_s / 4 eps0)(1 - q/q_s)^2, q = q_s t / (t + tau), tau = 4 eps0 / (rho mu)
Migration and efficiency. A charged grain moves relative to the gas at the Stokes velocity with Cunningham slip correction, w = q E C / (3 pi eta d), while the gas carries it along the tube and turbulence mixes it across the tube (a random walk with diffusivity D_t). With perfect mixing the fraction collected follows the Deutsch-Anderson law
eta = 1 - exp(- w A / Q) = 1 - exp(- 2 w L / (R v))
with A = 2 pi R L the collecting area and Q = pi R^2 v the gas flow. The page's particle simulation uses the local field, local ion density and growing charge of every grain; the Deutsch value with the wall-field w is shown for comparison: it assumes perfect mixing and the wall field everywhere, so the simulation can lie above it (grains near the wire feel a stronger field) or below it (weak mixing leaves a dust-poor layer at the wall).
Back corona. If the dust layer on the wall is very resistive, the ion current J = I / (2 pi R) drives a field E_layer = J rho_dust across it. Above about 1 MV/m the layer breaks down: the gap voltage drops by E_layer t (a 1 mm layer is assumed) and opposite-polarity ions neutralise the grains, which the model represents by a 60 % cut of the migration velocity.
Honest simplifications. (1) Only field charging is modelled; the diffusion charging that dominates below about 0.3 um is left out, so the efficiency for very fine dust is underestimated (real precipitators show a minimum around 0.1 to 1 um). (2) Negative corona with a constant ion mobility, no detail of the electron avalanche zone, no ionic wind, no particle space charge. (3) Plug flow with a uniform turbulent diffusivity that reflects at the wall (so only the electric drift collects dust); no boundary layer, no re-entrainment, no rapping. (4) Back corona is a threshold model with an assumed 60 % penalty and a fixed 1 mm layer. (5) Tube geometry only; real plant often uses plate-and-wire sections. (6) In the 3D view the tube is drawn 10 units in radius and 70 in length whatever the settings, and the ion dots drift faster than real for visibility.
Checks run in the page. Gauss's law integrated with the computed charge density; the solved field integrating to the gap voltage; the power balance integral of rho mu E^2 = I V; the current vanishing continuously at the Peek onset; the numerically integrated Pauthenier equation against its closed form; and a perfect-mixing particle simulation against the Deutsch-Anderson law.