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18-Env-A5 Air Quality and Pollution Control Engineering · December 2018

Question 2 of 5: Stack Plumes, Dispersion and the Gaussian Plume Model

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

18-Env-A5, Air Quality and Pollution Control Engineering — National Exam, December 2018. 3 hours, open book. The paper's notes state that Question 2 is compulsory and three (3) others complete a four-question paper; all five Problems are answered in full below.

Reference texts

Problem 2: Stack Plumes, Dispersion and the Gaussian Plume Model (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)a — simplified Gaussian plume equation. For a continuous point source of strength $Q$ at effective height $H$, with wind speed $u$ and horizontal/vertical dispersion coefficients $\sigma_y(x),\sigma_z(x)$, the ground-reflected concentration at downwind distance $x$, crosswind offset $y$ and height $z$ is

$$C(x,y,z)=\dfrac{Q}{2\pi u\,\sigma_y\sigma_z}\exp\!\left(\dfrac{-y^2}{2\sigma_y^2}\right)\left[\exp\!\left(\dfrac{-(z-H)^2}{2\sigma_z^2}\right)+\exp\!\left(\dfrac{-(z+H)^2}{2\sigma_z^2}\right)\right]$$

the second exponential term inside the brackets is the image source that models total reflection of the plume at the ground.

Part (i)b — significance of effective stack height. The effective stack height $H=h_s+\Delta h$ (physical stack height plus plume rise) is the height the Gaussian model treats as the point source; because $H$ enters the exponential term above as $(z-H)^2$, even a modest increase in $H$ produces a large drop in ground-level concentration directly beneath and near the plume — it is the single most powerful lever a stack designer has over near-field impact. One factor contributing to plume rise $\Delta h$: buoyancy rise — the flue gas leaves the stack hotter (and therefore less dense) than the ambient air, so it continues to rise under its own buoyancy well above the stack exit before bending over into the wind.

Part (i)c — effect of temperature and velocity on plume height. A larger exit-to-ambient temperature difference $\Delta T$ increases the buoyancy flux, so the plume rises higher and further before the ambient turbulence overtakes its momentum (this is the dominant term for hot utility-boiler stacks). A higher exit velocity increases the plume's initial vertical momentum, giving it more upward momentum rise before it bends over into the horizontal wind — this dominates for cooler process-vent stacks with little buoyancy of their own. Both effects raise the effective stack height $H$ and therefore lower the ground-level concentration predicted in part (a).

Part (ii) — four distinct plume behaviours. Coning, fanning, lofting and fumigation are chosen as a set that spans the full range of near-surface atmospheric stability, from best (lofting) to worst (fumigation) near-stack ground-level impact.

Coning (near-neutral)near-neutral lapse rate (ELR ≈ DALR) — overcast, moderate wind
Coning: a near-neutral lapse rate (environmental lapse rate ≈ dry adiabatic lapse rate), typical of overcast skies with moderate wind, gives roughly equal vertical and horizontal spreading — the classic textbook cone. Ground contact occurs gradually with distance, giving moderate, fairly predictable concentrations.
Fanning (stable inversion)stable layer through full plume depth — negligible vertical mixing
Fanning: a stable inversion spans the plume's full depth (typically a clear, calm night). Vertical eddies are suppressed almost entirely, so the plume spreads only horizontally into a thin, flat ribbon and can drift for very long distances without reaching the ground — a problem if it later strikes rising terrain or the inversion breaks and mixes suddenly downward.
Lofting (stable below, unstable aloft)inversion topunstable air aloft disperses upward freely; the inversion top caps downward mixing — no ground contact near the stack
Lofting: a stable layer sits below the plume (often the tail end of a nocturnal inversion) while the air above is unstable and disperses freely upward. This is the most favourable case — the stable layer below acts as a lid that prevents any downward mixing to grade near the stack, so ground-level concentrations stay very low until the surface inversion eventually erodes.
Fumigation (stable aloft, unstable below)inversion aloftplume trapped under the inversion breaks down abruptly as surface heating erodes the layer — rapid downward mixing to grade
Fumigation: the plume is initially trapped and fans out beneath an elevated inversion, but as morning surface heating erodes that inversion from below, the whole trapped mass mixes rapidly down to grade over a short distance — producing the worst short-term ground-level concentrations of any stability class, concentrated within a fairly narrow downwind band.

Potential dispersion problems, by type: coning gives steady, moderate concentrations that are the easiest to model and permit; fanning causes no near-field impact but can transport pollutant intact to distant, unexpected receptors (elevated terrain, or a delayed morning fumigation event); lofting is the safest operating condition, so it is often the target for scheduling high-emission operations; fumigation produces the highest transient ground-level concentrations recorded for any stability class and is the condition of greatest regulatory concern for morning start-up periods at power plants and refineries.

Part (iii) — Gaussian vs. Lagrangian dispersion models. The Gaussian model is a steady-state, analytical solution to the atmospheric diffusion equation that assumes a fixed wind field and normally-distributed concentration profiles about the plume centreline (parameterised by empirical $\sigma_y,\sigma_z$ curves for each Pasquill–Gifford stability class); it is fast, well-validated for flat terrain and near-field single sources, but assumes homogeneous, time-invariant meteorology over the whole plume travel path. The Lagrangian model tracks the trajectories of a large number of individual air parcels (or particles) as they are advected and randomly perturbed by a time- and space-varying turbulent wind field, then reconstructs the concentration field statistically from the parcel positions; it can represent calm winds, complex/varying terrain, non-stationary meteorology, and chemically reacting pollutants far more realistically than the Gaussian formulation, at substantially greater computational cost.

Check: this Problem is entirely qualitative — part (a)'s derivation manipulations shown are for the general form; no numeric substitution is required or possible from the given data.