Question 2 of 6: Air Quality Models and Gaussian Dispersion

Problem 2: Air Quality Models and Gaussian Dispersion (25 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.

Part (i) — three types of air quality models. (1) Gaussian plume models (e.g. AERMOD, ISC) represent the time-averaged concentration field from a steady point source as an analytical normal-distribution function of downwind distance, parameterised by atmospheric stability class; they assume a spatially uniform, statistically steady wind field. (2) Lagrangian (puff/particle) models (e.g. CALPUFF) release discrete parcels or puffs that are individually advected through a time- and space-varying 3-D wind field, building the concentration field from where the parcels travel; they handle calm and variable winds, complex terrain, and time-varying emissions that Gaussian models cannot. (3) Eulerian (grid) models (e.g. CMAQ, CAMx) solve the full 3-D advection–diffusion–reaction equation on a fixed spatial grid over an entire airshed, tracking chemical transformation (e.g. ozone photochemistry) between many interacting species simultaneously; they are the most computationally demanding but the only category that captures regional, multi-pollutant secondary chemistry. The key differences are governing assumption (steady analytical vs. Lagrangian trajectories vs. Eulerian fixed-grid transport), spatial/temporal flexibility, and whether atmospheric chemistry is represented at all.

Part (ii) — applying the Gaussian plume model. The Gaussian plume model is most appropriate for a single, continuously emitting point source over flat or gently rolling terrain, under steady meteorological conditions — the classic case is a screening-level environmental assessment for a new industrial stack (e.g. a proposed cement kiln or power plant) where the regulator needs a fast, conservative estimate of maximum ground-level concentration for a permit application. Assumptions that must hold: (1) the wind field is spatially uniform and temporally steady over the averaging period (typically one hour); (2) the plume's mass is conserved (no chemical reaction or deposition along the way, or these are added as simple decay/depletion terms); (3) the terrain is flat and free of buildings that would cause downwash or channel the plume; (4) emissions are continuous and constant over the averaging period, not an instantaneous puff release. Real case: a proposed 500 t/d coal-fired boiler in a flat river valley uses AERMOD (a refined Gaussian model) to demonstrate that its worst-case 1-hour ground-level SO₂ concentration meets the CCME/CAAQS standard before a permit is issued — exactly the screening role Gaussian models are built for.

Part (iii) — solar radiation, wind fields and a third aspect on stack plumes. Solar radiation heats the ground during the day, driving a superadiabatic (unstable) lapse rate that generates large convective eddies; in the Gaussian framework this manifests as large $\sigma_z$ (rapid vertical plume growth), which dilutes the plume quickly but can also cause "looping," where the plume periodically touches the ground close to the stack with brief, intense peak concentrations. At night, radiative cooling produces a stable inversion (small $\sigma_z$), suppressing vertical mixing and causing "fanning," a thin ribbon that drifts far downwind without reaching the ground. Wind field (speed $u$) enters directly in the denominator of $C=\dfrac{Q}{\pi\sigma_y\sigma_z u}(\cdots)$: concentration is inversely proportional to wind speed, so a calm wind gives the highest ground-level concentrations (and near-zero $u$ invalidates the steady-state Gaussian assumption altogether), while a strong wind dilutes the plume quickly but also bends it over sooner, reducing effective stack height. A third aspect, atmospheric stability class (Pasquill–Gifford), sets the functional form of $\sigma_y(x)$ and $\sigma_z(x)$ directly: an unstable class (A/B) grows both dispersion coefficients quickly with distance (rapid dilution, but also fumigation risk when a morning inversion breaks), while a stable class (E/F) grows them very slowly, producing long, narrow, poorly diluted plumes that can travel tens of kilometres before reaching ground level.