Question 1 of 7: Gaussian Plume Dispersion and Plume Behaviour

Problem 1: Gaussian Plume Dispersion and Plume Behaviour (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.

Part (i)a — simplifying to the maximum ground-level concentration.

Given. The exam's own Gaussian plume equation for concentration $C_x$ at downwind distance $x$, crosswind offset $y$ and height $z$, for a source of strength $Q$, effective stack height $H$, wind speed $u$, and dispersion coefficients $\sigma_y(x)$, $\sigma_z(x)$ (the second exponential term is the ground-reflected image source).

Find. The simplified closed-form expression for the maximum ground-level concentration $C_{max}$ and the condition under which it occurs.

Approach. Evaluate the plume centreline at ground level ($y=0$, $z=0$), then find the downwind distance (expressed through $\sigma_z$) at which this centreline, ground-level concentration is largest.

1. Ground level, plume centreline. Set $y=0$ (centreline) and $z=0$ (ground). The crosswind exponential becomes 1, and both image-source exponentials become identical: $$\exp\!\left(\dfrac{-(0-H)^2}{2\sigma_z^2}\right)+\exp\!\left(\dfrac{-(0+H)^2}{2\sigma_z^2}\right)=2\exp\!\left(\dfrac{-H^2}{2\sigma_z^2}\right)$$ so $$C(x,0,0)=\dfrac{2Q}{\pi\,\sigma_y\sigma_z\,u}\exp\!\left(\dfrac{-H^2}{2\sigma_z^2}\right)$$
2. Introduce the near-constant stability ratio. Both $\sigma_y$ and $\sigma_z$ grow with $x$ under a given Pasquill–Gifford stability class; over the range that matters here their ratio $p=\sigma_z/\sigma_y$ is approximately constant, so $C$ can be treated as a function of $\sigma_z$ alone via $\sigma_y=\sigma_z/p$: $$C(\sigma_z)=\dfrac{2Qp}{\pi u\,\sigma_z^{2}}\exp\!\left(\dfrac{-H^2}{2\sigma_z^{2}}\right)$$
3. Maximise with respect to $\sigma_z$. Setting $dC/d\sigma_z=0$ (equivalently $d/d(\sigma_z^2)=0$) gives the classical stationary-point condition $$\boxed{\sigma_z=\dfrac{H}{\sqrt{2}}}$$ i.e. the maximum ground-level concentration occurs at the downwind distance where the vertical spread has grown to $H/\sqrt2$.
4. Substitute back. At $\sigma_z^2=H^2/2$, $\exp(-H^2/2\sigma_z^2)=\exp(-1)=1/e$, giving $$\boxed{C_{max}=\dfrac{4Q}{\pi\, e\, u\, H^{2}}\left(\dfrac{\sigma_z}{\sigma_y}\right)}$$

Part (i)b — significance of effective stack height. The effective stack height $H=h_s+\Delta h$ is the height at which the plume centreline effectively behaves as a point source, not the physical stack height $h_s$ alone; it appears squared in $C_{max}$ above, so even a modest increase in $H$ produces a large drop in ground-level concentration — it is the single most powerful lever a designer has over downwind impact. Two factors that contribute to the plume rise $\Delta h$: (1) momentum (velocity) rise — the exit gas leaves the stack with vertical velocity and carries upward momentum before it bends over into the wind; (2) buoyancy rise — the exhaust is warmer (less dense) than the ambient air, so it continues to rise as a thermal plume until it entrains enough cool air to lose its buoyancy, per Briggs' plume-rise formulas.

Part (i)c — effect of temperature and velocity on plume height. A higher stack-gas exit velocity increases momentum rise and also physically projects the plume further above the stack lip before it is bent over by the crosswind, so plume rise increases roughly with exit velocity. A larger exit-to-ambient temperature difference increases the initial buoyancy flux, driving greater thermal rise; conversely, on a well-insulated or heat-recovered stack the exhaust is cooler and the plume rises less, so a larger $H$ (and hence the $C_{max}\propto 1/H^2$ benefit) depends on keeping the flue gas hot and fast rather than throttling it for efficiency alone.

Part (ii) — two plume behaviours. Looping and Fanning are chosen as a deliberately contrasting pair: one is the atmospheric condition of best dilution but worst instantaneous peaks, the other of poorest dilution but no ground contact.

Potential dispersion problems: looping's brief, intense ground-level spikes near the stack are hard for continuous monitors to average out and can trip short-term (1-hour) air-quality standards even though the daily average is low; fanning gives excellent short-term ground-level air quality but stores pollutant aloft, and if the inversion later erodes (fumigation) or the ribbon strikes elevated terrain, the accumulated mass reaches receptors all at once.

Part (iii) — Gaussian vs. Lagrangian dispersion models. A Gaussian model assumes a statistically stationary, spatially uniform wind field and represents the time-averaged plume as a fixed analytical (normal-distribution) concentration field anchored to the source, parameterised only by $\sigma_y(x)$, $\sigma_z(x)$ for a chosen stability class — it is fast, well validated for flat terrain and steady conditions, but cannot represent calm/variable winds, complex terrain, or chemically reacting plumes without empirical patches. A Lagrangian (particle- or puff-following) model instead releases a large number of discrete parcels (or puffs) and tracks each one's trajectory through a time- and space-varying wind and turbulence field, building the concentration field from where the parcels end up; this handles calm winds, terrain-channelled flow, time-varying emissions, and in-plume chemistry naturally, at the cost of much greater computational effort and the need for a full 3-D meteorological wind/turbulence field as input, whereas Gaussian models need only a handful of summary statistics (wind speed, stability class).