Question 5 of 7: Atmospheric Stability, Dispersion Modelling and the Gaussian Plume Equation

Question 5: Atmospheric Stability, Dispersion Modelling and the Gaussian Plume Equation (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) Pasquill–Gifford Class A (Extremely Unstable) Conditions

Class A occurs on clear, sunny days with strong solar insolation and light surface wind (typically <2 m/s), which together maximize surface heating and drive strong convective (thermal) turbulence in the boundary layer — the intense solar radiation heats the ground, which in turn heats the air immediately above it, and with little wind to mechanically mix or advect that heat away, large buoyant thermals rise and dominate the turbulence field. This strong vertical mixing produces "looping" plume behaviour: the plume is carried in large, meandering vertical excursions by the convective eddies rather than spreading smoothly, giving very effective overall (time-averaged) dilution but a risk of brief, intermittent high ground-level concentrations when a loop happens to bring the plume core down to grade near the stack. Class A conditions are predicted from the standard Pasquill–Gifford lookup table using two inputs alone — surface wind speed and incoming solar radiation intensity (or, at night, cloud cover) — so a forecast of strong sun and calm wind is sufficient to anticipate Class A dispersion without any turbulence measurement.

(ii) Dispersion/Deposition Modelling: One Model, Two Assumptions, Two Limitations

Dispersion and deposition models translate a known or estimated emission rate into a predicted spatial and temporal pattern of ground-level concentration (and, for deposition models, cumulative surface loading), which lets an engineer or regulator assess compliance with an ambient air-quality standard, site a new stack, or evaluate a proposed control strategy before it is built — without a model, the only way to know the ground-level impact of a source would be to build it first and measure.

Model: the steady-state Gaussian plume model (used in Question 5(iii)), which assumes the pollutant concentration is normally (Gaussian) distributed in the crosswind (y) and vertical (z) directions about the plume centerline, with the spread parameters σ_y and σ_z growing with downwind distance according to the prevailing atmospheric stability class.

Two key assumptions. (1) Steady-state emission rate and constant, uniform meteorology (wind speed, direction and stability class) over the travel time from source to the receptor. (2) Conservation of pollutant mass within the plume — no chemical transformation, deposition removal or ground absorption is occurring unless separately superimposed on the base equation (e.g., a reflection term at the ground or a decay term).

Two limitations. (1) The model assumes flat, homogeneous terrain and an unobstructed wind field, so it performs poorly in complex terrain, urban street canyons, or near large buildings where wake and downwash effects dominate. (2) The steady-state assumption breaks down for calm winds (u→0, where the equation is undefined), very stable conditions with intermittent fumigation, or long transport distances/times over which the meteorology genuinely changes — conditions common at night and in valleys, precisely when some of the highest ground-level concentrations actually occur.

(iii) Off-Centerline SO₂ Concentration from the Gaussian Plume Equation

Given. A point-source Gaussian plume at $x=1\,\text{km}$ downstream:

Find. The SO₂ concentration $C(x,y,z)$ at the point 60 m to the side and 20 m below the plume centerline.

Approach. Substitute the given values directly into the supplied Gaussian plume equation, evaluating the crosswind and vertical exponential terms separately before combining them.

1. Crosswind term. $\dfrac{y^2}{\sigma_y^2} = \dfrac{60^2}{30^2} = \dfrac{3600}{900} = 4.00$.
2. Vertical term. $\dfrac{(z-H)^2}{\sigma_z^2} = \dfrac{(-20)^2}{20^2} = \dfrac{400}{400} = 1.00$.
3. Exponent and exponential factor. $-\dfrac{1}{2}(4.00+1.00) = -2.50$, so $\exp(-2.50) = 0.0821$.
4. Pre-exponential (peak centerline-plane) factor. $\dfrac{Q}{2\pi u \sigma_y \sigma_z} = \dfrac{20}{2\pi(5)(30)(20)} = \dfrac{20}{18{,}850} = 1.061\times10^{-3}\ \text{g/m}^3$.
5. Combine. $$C = \left(1.061\times10^{-3}\right)(0.0821) = \boxed{8.71\times10^{-5}\ \text{g/m}^3 \approx 87.1\ \mu\text{g/m}^3}.$$