Stack Plumes, Dispersion and the Gaussian Plume Model: 18-Env-A5 Air Quality and Pollution Control Engineering, Undated paper, Q2

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. The printed equation already includes both the real source term and its ground-reflected image (the two terms inside the braces). The useful simplification for design purposes is the concentration at ground level on the plume centreline ($y=0$, $z=0$) — directly beneath and downwind of the stack, where the two exponential terms in the braces become identical:

$$C(x,0,0)=\dfrac{2Q}{\pi\,u\,\sigma_y\sigma_z}\exp\!\left(\dfrac{-H^2}{2\sigma_z^2}\right)$$

This single expression is what a stack designer actually evaluates repeatedly (at successive downwind distances $x$, through the $x$-dependence of $\sigma_y(x),\sigma_z(x)$) to find the worst-case ground-level concentration and the distance at which it occurs.

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 above as $H^2$ in the denominator's exponent, 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, more effective than adding downstream flue-gas treatment for reducing local ground-level exposure alone. 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 physical 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 ambient turbulence overtakes its momentum — this dominates 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); conversely a low exit velocity or a plume close to ambient temperature gives little rise, so $H\approx h_s$ and the plume behaves as though released almost at the physical stack top.

Part (ii) — four distinct plume behaviours. Looping, coning, fanning and trapping are chosen as a set spanning the full range of near-surface atmospheric stability from most turbulent (looping) to a fully-capped stable layer (trapping), each drawn as a side-view profile with the stack on the left and downwind distance increasing to the right.

Looping: strongly unstable daytime conditions (steep superadiabatic lapse rate, strong solar heating) produce large convective eddies that loop the plume centreline up and down. Dispersion is fastest of any class (lowest long-term average concentration), but the instantaneous ground-level concentration directly under a downward loop can briefly spike very high — a short-duration, high-intensity impact problem rather than a sustained one.

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 — the easiest stability class to model and permit against.

Fanning: a stable inversion (typically a clear, calm night) spans the plume's full depth. 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.

Trapping: the plume is released into a shallow unstable/neutral layer that is itself capped by an inversion aloft (commonly a subsidence inversion under a stagnant high-pressure system), while the ground remains near-neutral to slightly stable below. Both lids prevent the plume from escaping vertically, so pollutant accumulates and concentration builds steadily with time — the classic multi-day air-stagnation episode.

Potential dispersion problems by type: looping's brief high-intensity downward gusts make short-term (peak, not average) ground-level standards the binding constraint; fanning transports pollutant intact to distant, unexpected receptors and is dangerous if the inversion later breaks (delayed fumigation); coning gives steady, moderate, easily-modelled concentrations; trapping accumulates pollutant over successive days under a stagnant high-pressure system, historically the meteorological signature behind the worst urban air-pollution episodes (e.g. the 1952 London smog).

Part (iii) — Gaussian vs. Lagrangian dispersion models. 1. Underlying formulation. The Gaussian model is a steady-state, closed-form analytical solution to the atmospheric diffusion equation that assumes a fixed, uniform 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. The Lagrangian model instead 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 accumulated parcel positions — no closed-form profile is assumed. 2. Applicability. The Gaussian approach is fast and well-validated for flat terrain, near-field single sources, and steady meteorology, but breaks down under calm/light winds (the $1/u$ term in the equation is singular as $u\to0$), complex terrain, or non-stationary conditions. The Lagrangian approach handles calm winds, complex/varying terrain, non-stationary meteorology and chemically reacting or multi-source pollutant fields far more realistically, at substantially greater computational cost.

Check: this Problem is entirely qualitative and symbolic — part (a)'s reduction is a substitution ($y=0,z=0$) into the general form printed on the exam's own page; no numeric substitution is required or possible from the given data.

Topic: Gaussian plume dispersion and atmospheric stability classes.

Key relations: Gaussian plume equation with ground reflection; ground-level centreline reduction $C(x,0,0)=2Q/(\pi u\sigma_y\sigma_z)\exp(-H^2/2\sigma_z^2)$; effective stack height $H=h_s+\Delta h$; Pasquill–Gifford stability classes A–F and the six classical plume shapes (looping, coning, fanning, lofting, fumigation, trapping).

Why this works: plume shape is a direct, visual readout of the vertical temperature (lapse-rate) structure of the boundary layer at stack height — comparing the environmental lapse rate to the dry adiabatic lapse rate at each level tells you whether a parcel displaced from that level accelerates away (unstable) or is restored (stable), which is exactly what governs vertical mixing and hence which of the six shapes results.

Common pitfall: assuming a larger, more visually dramatic plume envelope always means worse ground-level impact — fanning has a very large lateral/downwind extent but essentially zero near-field ground contact, while looping's compact excursions can still deliver a brief high peak concentration.

Source: Wark, Warner & Davis, Air Pollution: Its Origin and Control, 3rd ed., Ch. 11 (Atmospheric Dispersion).

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

Problem 2: Stack Plumes, Dispersion and the Gaussian Plume Model (25 marks)