Question 2 of 7: Gaussian Plume Dispersion Model

Question 2: Gaussian Plume Dispersion Model (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)(a) Simplifying to the Maximum Ground-Level Concentration

Approach. Ground level means $z=0$; the plume centreline means $y=0$ (the two locations combine to give the single largest concentration for a given downwind distance $x$); "maximum" further means finding the downwind distance $x_{max}$ at which this ground-level centreline value peaks, since $\sigma_y,\sigma_z$ both grow with $x$ while the exponential decays.

1. Set $y=0$ (centreline). The crosswind exponential $\exp(-y^2/2\sigma_y^2)\to 1$, leaving $$C(x,0,z,H)=\frac{Q}{2\pi u\sigma_y\sigma_z}\left[\exp\!\left(\frac{-(z-H)^2}{2\sigma_z^2}\right)+\exp\!\left(\frac{-(z+H)^2}{2\sigma_z^2}\right)\right].$$
2. Set $z=0$ (ground level). Both bracket terms become identical, $\exp(-H^2/2\sigma_z^2)$, and add: $$C(x,0,0,H)=\boxed{\dfrac{Q}{\pi u\sigma_y\sigma_z}\exp\!\left(\dfrac{-H^2}{2\sigma_z^2}\right)}.$$ This is the standard ground-level centreline concentration — already the working form used for "where is the plume worst at ground level."
3. Maximize over $x$. Differentiating $C(x,0,0,H)$ with respect to $x$ (through $\sigma_y(x),\sigma_z(x)$) and setting $dC/dx=0$ gives the classical condition $$\sigma_z(x_{max})=\frac{H}{\sqrt2},$$ i.e. the peak ground-level concentration occurs at the downwind distance where the vertical plume spread has grown to $H/\sqrt2$. Substituting back, $$C_{max}=\boxed{\dfrac{2Q}{\pi e\,u\,H^2}\left(\dfrac{\sigma_z}{\sigma_y}\right)_{x=x_{max}}}.$$

(i)(b) Effective Stack Height

The effective stack height $H=h_s+\Delta h$ is the physical stack height $h_s$ plus the plume rise $\Delta h$ the effluent gains after leaving the stack mouth before it behaves as a passive Gaussian puff. It matters because $C_{max}$ above is inversely proportional to $H^2$ — even a modest increase in effective height dramatically lowers ground-level concentrations near the source. Two contributing factors: (1) momentum — the effluent leaves the stack with a vertical exit velocity, carrying it upward as a buoyant jet before ambient turbulence bends it over; (2) buoyancy — combustion gases are hotter (less dense) than the surrounding air, so the plume continues to rise thermally after momentum is dissipated, an effect that is stronger on unstable days and weaker under a strong inversion.

(i)(c) Two Assumptions Behind the Gaussian Plume Model

(1) Steady, uniform wind and stationary turbulence. The model assumes a constant wind speed and direction, and horizontally homogeneous turbulence intensity (constant $\sigma_y,\sigma_z$ growth) over the travel time from source to receptor. Example: it works well for a tall stack in flat, open terrain under steady synoptic winds, but breaks down under calm/variable winds or complex terrain where the wind field itself changes along the plume's path.

(2) Conservative pollutant with total ground reflection. The pollutant is assumed chemically inert (no reaction, deposition or decay along the way), and the ground is modelled as a perfect reflecting boundary (the "$+H$" image-source term), conserving all emitted mass in the plume. Example: reasonably valid for CO or SO₂ over short travel times; poor for a highly reactive or rapidly depositing species such as ozone precursors undergoing photochemistry, or a sticky/soluble gas that deposits efficiently on vegetation or wet surfaces.

(i)(d) Night-time vs. Midday Plume Behaviour

At night, the ground radiates heat away faster than the air aloft, producing a surface-based temperature inversion (a stable atmosphere, Pasquill class E–F) with zero solar insolation and essentially no thermal turbulence. Vertical mixing is strongly suppressed, so the plume disperses mainly horizontally, travelling far downwind as a thin, only slightly diluted ribbon (a "fanning" plume — see part (ii)).

By lunch time, strong solar insolation has heated the ground, producing a superadiabatic lapse rate (an unstable atmosphere, Pasquill class A–B) with vigorous convective turbulence. The same plume now mixes rapidly in large eddies both above and below its centreline (a "looping" plume — see part (ii)), diluting much faster overall but occasionally bringing a high-concentration eddy down to ground level close to the stack.

(ii) Two Plume Behaviours — Fanning and Looping

Fanning occurs under a strong surface-based inversion (typically night-time/early morning, Pasquill F). Vertical spread is almost completely suppressed while horizontal (crosswind) spread continues normally, so in side view the plume is a thin, nearly flat ribbon that barely widens with distance from the stack.

Potential problem: because vertical mixing is suppressed, the plume can travel long distances at nearly undiluted concentration; if it intercepts elevated terrain, or if the inversion breaks up suddenly at sunrise and mixes the accumulated plume rapidly to the ground, high ground-level concentrations can occur well downwind of the source.

Looping occurs under strongly unstable, convective conditions (typically sunny midday, Pasquill A–B). Large thermal eddies loop the plume up and down about its mean centreline with growing amplitude as it travels downwind.

Potential problem: overall dilution is fast (large $\sigma_y,\sigma_z$), but individual convective eddies can intermittently bring a concentrated parcel of the plume down to ground level quite close to the source, producing brief, high, unpredictable ground-level concentration spikes (a nuisance/odour-complaint pattern rather than a sustained exposure).

(iii) Dispersion and Deposition Modelling — Eulerian and Lagrangian Approaches

Eulerian (fixed-grid) models solve the pollutant advection-diffusion equation on a fixed three-dimensional grid in space, tracking how concentration in each grid cell changes over time due to advection, turbulent diffusion, chemical transformation and deposition. Because the whole domain is resolved simultaneously, Eulerian models are well suited to regional-scale problems where chemistry between many reacting species matters, such as photochemical smog/ozone models (e.g. CMAQ-type regional air-quality models) covering an entire airshed.

Lagrangian (particle/puff) models instead track the trajectories of many discrete pollutant parcels ("puffs" or particles) as each is advected by the local wind field and spread by turbulence, following the fluid's own moving reference frame rather than a fixed grid. This makes them well suited to a single or moving source, complex/varying terrain, and time-varying meteorology, since each puff's dispersion parameters can be updated individually along its own path (e.g. CALPUFF-type puff models for a single industrial source in complex terrain).