Question 3 of 7: Gaussian and Puff Dispersion Models and Effective Stack Height

Problem 3: Gaussian and Puff Dispersion Models and Effective Stack Height (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) — Gaussian/Eddy vs. Puff models. (1) Steady vs. unsteady emissions: the Gaussian/Eddy plume model assumes a continuous, steady emission rate $Q$ and constant meteorology over the travel time to the receptor, giving one stationary concentration field; a Puff model represents the release as a sequence of discrete puffs, each advected and dispersed independently, so it naturally handles unsteady/intermittent releases and time-varying meteorology. (2) Wind field treatment: the Gaussian model requires a single, spatially uniform wind vector for the whole domain; Puff models can be driven by a full 3-D, time-varying wind field, making them suitable for complex terrain, coastal/sea-breeze circulations and calm or highly variable wind. (3) Low/calm wind behaviour: the Gaussian equation has $u$ in the denominator and becomes singular (undefined/unrealistically high concentration) as $u\rightarrow 0$; Puff models remain well-behaved at calm wind since each puff simply disperses radially about its release point rather than being advected downwind, making them the appropriate choice for worst-case screening under stagnant, high-stability conditions — often the most polluted episodes.

Part (ii) — assumptions/approximations of the Gaussian model. (1) Steady-state emission and meteorology: $Q$, $u$ and wind direction are all taken as constant for the duration of pollutant travel from source to receptor. (2) Conservative pollutant: no chemical reaction, decay or deposition removes mass from the plume as it travels (the full emitted mass is conserved within the Gaussian envelope). (3) Gaussian (normal) concentration distribution in both the crosswind ($y$) and vertical ($z$) directions, with the dispersion coefficients $\sigma_y,\sigma_z$ functions only of downwind distance $x$ and the atmospheric stability class (Pasquill–Gifford curves), independent of the pollutant itself. Engineering implication: because the model over-predicts or becomes undefined as $u\rightarrow 0$ and cannot represent a reacting/depositing pollutant, engineers apply a minimum design wind speed (typically $u\ge 1\text{-}2\ \text{m/s}$) when using the Gaussian equation for regulatory worst-case assessment, and switch to a puff or reactive-transport model for calm-wind or chemically active pollutant scenarios.

Part (iii) — effective stack height. The effective stack height $H$ is the elevation the plume centreline actually reaches above ground, once the buoyant/momentum-driven rise of the hot, high-velocity exhaust above the physical stack top is added: $H = H_s + \Delta h$, where $H_s$ is the physical stack height and $\Delta h$ is the plume rise. A typical plume-rise formula (Holland's equation) is $$\Delta h = \frac{v_s d_s}{u}\left(1.5 + 2.68\times10^{-3}\,P\,\frac{T_s-T_a}{T_s}\,d_s\right)$$ where $v_s$ is the stack exit gas velocity, $d_s$ the stack diameter, $u$ the wind speed, $P$ the atmospheric pressure (mb), and $T_s,T_a$ the stack-gas and ambient temperatures (K). Two variables that affect $\Delta h$: (1) stack exit velocity $v_s$ — a higher exit velocity increases the momentum flux carrying the plume upward, directly increasing $\Delta h$; (2) the temperature difference $T_s-T_a$ (buoyancy) — a hotter effluent relative to ambient air is more buoyant and rises further before cooling to ambient density, so $\Delta h$ increases with $T_s-T_a$ (conversely, higher wind speed $u$ in the denominator reduces $\Delta h$ by shortening the plume's residence time to rise).