Question 4 of 7: Stack Plume Behaviour and Gaussian/Eddy Dispersion Modelling

Question 4: Stack Plume Behaviour and Gaussian/Eddy Dispersion Modelling (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) Three Primary Plume Types

The three classic plume shapes correspond to the three broad atmospheric stability regimes, driven by how solar radiation and wind together set the vertical temperature (lapse-rate) profile. A fanning plume occurs under stable conditions — a clear night with light wind and a temperature inversion actively suppresses vertical motion, so the plume spreads almost only horizontally in a thin ribbon at stack height, with very low ground-level concentration directly beneath it. A looping plume occurs under very unstable conditions — strong daytime solar heating drives vigorous convective eddies that swing the plume up and down erratically, occasionally bringing a concentrated pocket of pollutant to the ground close to the source before it lifts away again. A coning plume occurs under near-neutral or slightly unstable conditions with moderate mixing and no violent looping — a steadily widening cone downwind, the well-behaved baseline case the standard Gaussian equation is built around.

(ii) Modelling Dispersion/Deposition and the Measurements Needed to Verify It

A Gaussian plume model assumes the pollutant concentration downwind of a continuous point source follows a normal (Gaussian) distribution in both the crosswind ($y$) and vertical ($z$) directions, with the spread of each distribution parameterized by dispersion coefficients $\sigma_y(x)$ and $\sigma_z(x)$ that grow with downwind distance $x$ according to empirical curves keyed to the Pasquill–Gifford atmospheric stability class (itself set by solar radiation, cloud cover and wind speed). Dry deposition is layered onto this by applying a deposition velocity $v_d$ at the ground boundary (a depletion term that removes mass from the plume as it advects downwind), while wet deposition (washout/rainout) is modelled as an additional first-order scavenging-coefficient loss term active only during precipitation. An Eddy-diffusion (K-theory) model instead solves the advection–diffusion equation directly with an eddy diffusivity $K$ representing turbulent mixing — more physically general (it can handle complex terrain and time-varying meteorology that a steady-state Gaussian formula cannot) but computationally heavier and requiring a turbulence closure scheme.

Verifying either model requires field measurements that close the loop between the model's inputs and its predicted output: (1) meteorological data — wind speed and direction, atmospheric stability class (from turbulence intensity, solar radiation, or a sigma-theta wind-direction-fluctuation measurement), and mixing-height, all needed as model inputs; and (2) ambient ground-level concentration measurements at multiple downwind receptor locations (continuous analyzers or passive samplers, Question 5), compared statistically against the model's predicted concentration field to check both the peak magnitude and its downwind location — a model that gets $C_{max}$ right but predicts it at the wrong distance is still not validated.

(iii) Three Limitations of the Steady-State Gaussian/Eddy Model

(1) Steady-state assumption. The model assumes constant emission rate and unchanging meteorology (wind speed/direction, stability) over the travel time from source to receptor; it therefore performs poorly during rapidly changing conditions (frontal passage, diurnal transition, calm/light-and-variable wind), which are also frequently the conditions associated with the highest real ground-level concentrations.

(2) Flat, homogeneous terrain assumption. The Gaussian formula was empirically derived over flat, open terrain; complex terrain (valleys, coastal shorelines, urban street canyons) distorts the actual flow field and mixing in ways the simple $\sigma_y,\sigma_z$ curves do not capture, requiring terrain-adjustment algorithms or a full 3-D Eddy/CFD model instead.

(3) Poor performance at very low wind speed / near the source. The concentration formula has $u$ in the denominator, so as wind speed approaches zero the predicted concentration becomes unphysically large (or undefined); near-calm conditions must be handled with a separate calm-wind sub-model, and very close to the stack (within a few stack heights) the plume-rise and initial-dilution physics are not yet well represented by the far-field Gaussian form.