Question 5 of 7: Dispersion Behaviour of Gaseous Pollutants and Particulate Control

Question 5: Dispersion Behaviour of Gaseous Pollutants and Particulate Control (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(i) Temperature, Stability and Mixing Height

The ambient temperature profile (the environmental lapse rate) sets atmospheric stability: comparing the actual lapse rate to the dry adiabatic lapse rate ($\approx 9.8\ {}^{\circ}\text{C/km}$) determines whether a displaced air parcel keeps rising (unstable — environmental lapse rate steeper than adiabatic, strong vertical mixing) or sinks back (stable — environmental lapse rate shallower than adiabatic, or an actual temperature inversion where temperature increases with height, which suppresses vertical mixing almost entirely). This is exactly the physical basis of the Pasquill classes already discussed in Question 2: unstable conditions give fast-growing $\sigma_y(x)$, $\sigma_z(x)$ (rapid dilution, but a looping plume); stable conditions give slow-growing $\sigma$'s (a coherent plume with potential long-range transport).

Mixing height, $L$, is the depth of the well-mixed surface layer capped by an elevated inversion (either a subsidence inversion or the top of the daytime convective boundary layer); it defines the vertical air volume available to dilute pollutants emitted near the surface. A shallow mixing height — common on clear, calm winter nights or in a valley/basin airshed under a persistent inversion — confines pollutants to a thin layer and raises ground-level concentration for a given emission rate; a deep, well-developed mixing height (a sunny, unstable afternoon) dilutes through a much larger volume and lowers concentration. Once the Gaussian plume's $\sigma_z$ approaches roughly $L/2.15$, the plume is effectively trapped between the ground and the inversion lid (see figure below); the model then switches to a limited-mixing form — an infinite series of image-source reflections off both the ground and the lid — that eventually produces a uniform vertical concentration profile, $$C = \frac{Q}{\sqrt{2\pi}\,\sigma_y\, u\, L},$$ rather than the unbounded Gaussian vertical decay. The worst ground-level concentrations occur from the combination of a low mixing height, stable/near-calm conditions and, for an elevated source, fumigation — pollutant accumulated aloft overnight in a stable layer being rapidly mixed down to the surface as a morning inversion breaks up from below.

(ii) Cyclonic Collector Efficiency vs. Particle Size

A cyclone's collection efficiency increases with particle size, because larger particles experience a proportionally larger centrifugal force in the vortex relative to the aerodynamic drag resisting that motion. This behaviour is characterized by the cyclone's cut diameter, $d_{p50}$ — the particle diameter collected at 50% efficiency — obtained from a semi-empirical correlation (e.g. Lapple's) based on the cyclone's geometry, inlet velocity, gas viscosity, particle/gas density difference and effective number of vortex turns. Efficiency then follows a sigmoid-shaped grade-efficiency curve plotted against the ratio $d_p/d_{p50}$: efficiency approaches 0% for very fine particles (drag dominates; the particle simply follows the gas streamlines out the top) and approaches 100% for coarse particles (centrifugal force dominates; the particle is thrown to the wall and slides down into the dust hopper).

Because a single cyclone captures the fine fraction poorly no matter how it is sized, this size-dependent efficiency curve is used to design a multi-stage particulate control train rather than relying on the cyclone alone: the cyclone (cheap, robust, low maintenance) is sized as a coarse pre-cleaner that removes the bulk of the mass loading (the larger particles, well above its $d_{p50}$) ahead of a higher-efficiency final control device — a baghouse or electrostatic precipitator — that economically handles the fine fraction the cyclone cannot capture. This staged approach achieves efficient overall particulate reduction across the full size range at lower total system cost than trying to size a single cyclone (or a single device of any type) for the finest particles alone.

(iii) Deutsch–Anderson Equation — Electrostatic Precipitator

The Deutsch–Anderson equation for ESP collection efficiency is $$\eta = 1 - \exp\!\left(-\frac{w A}{Q}\right).$$

$w$ — migration (drift) velocity [m/s]: the effective velocity at which a charged particle migrates through the gas toward the collecting plate under the applied electric field. It depends on particle charge, field strength, particle size and gas viscosity; a larger, more strongly charged particle in a stronger field has a higher $w$ and is collected more efficiently for a given precipitator size.

$A$ — total collecting plate area [m²]: the surface area available for particles to migrate onto and be collected. This is the principal design/cost lever — more or longer plates directly increase $A$ and therefore efficiency for a fixed gas flow.

$Q$ — volumetric gas flow rate [m³/s]: for a fixed $A$ and $w$, a higher $Q$ means a shorter gas residence time within the electric field, leaving particles less time to migrate to a plate before being swept out — efficiency decreases as throughput increases. Together $A$ and $Q$ combine into the specific collection area $A/Q$, the single parameter engineers size an ESP around for a target removal efficiency.