Question 2 of 7: Plume Rise and the Gaussian Dispersion Model

Question 2: Plume Rise and the Gaussian 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) Plume Rise (Briggs Buoyant Plume Formula)

Given. A buoyant plume rises from a stack under unstable atmospheric conditions:

Find. The plume rise $\Delta h$ and the effective stack height $H = H_s+\Delta h$.

Approach. For unstable/neutral stability with $F<55\ \text{m}^4/\text{s}^3$, use Briggs' two-stage formula: first the downwind distance to final rise $x_f$, then the final plume rise $\Delta h$ itself.

1. Distance to final rise. For $F<55\ \text{m}^4/\text{s}^3$: $x_f = 49\,F^{0.625}$. $$x_f = 49(50)^{0.625} = \boxed{565.0\ \text{m}}.$$
2. Final plume rise. $\Delta h = \dfrac{1.6\,F^{1/3}\,x_f^{2/3}}{u}$. $$\Delta h = \frac{1.6(50)^{1/3}(565.0)^{2/3}}{5} = \boxed{80.6\ \text{m}}.$$
3. Effective stack height. $H = H_s + \Delta h = 40+80.6 = \boxed{120.6\ \text{m}}$.

(ii) Simplifications in Gaussian/Eddy Dispersion Modelling

Two key simplifications underlie both the Gaussian plume model and simpler Eddy (K-theory) diffusion treatments. First, the models assume steady-state, uniform meteorology: emission rate $Q$, wind speed $u$, wind direction and atmospheric stability class are all taken as constant over the travel time from source to receptor, so a single Pasquill–Gifford stability class fixes $\sigma_y$ and $\sigma_z$ for the whole plume rather than allowing them to evolve with a changing boundary layer. Second, the models assume no chemical or physical transformation and no net removal along the plume path — the pollutant mass is treated as conserved (Gaussian-distributed dispersion only), ignoring reaction (e.g. SO₂→sulfate), dry/wet deposition loss, and gravitational settling except where a separate deposition-velocity correction is bolted on afterward. Both simplifications trade physical completeness for a closed-form solution and are reasonable over the short travel times (minutes to a few hours) for which the models are normally applied, but they break down for long-range transport, complex/mountainous terrain, or highly reactive pollutant systems (e.g. photochemical smog, Question 6(iii)).

(iii) Minimizing Environmental Impact from Stack Emissions

The model shows $C_x \propto \frac{1}{\sigma_y\sigma_z u}\exp\!\left(\frac{-H^2}{2\sigma_z^2}\right)$, so ground-level concentration falls steeply as effective height $H$ increases and rises linearly with emission rate $Q$. Three efficient design responses follow directly: (1) increase the effective stack height $H=H_s+\Delta h$ — both physical stack height and plume buoyancy/momentum (Part (i)) push the plume centreline higher before it can touch ground, and because $C_x$ depends on $H^2$ in the exponent this is the single most effective lever; (2) reduce the emission rate $Q$ at the source (better combustion control, fuel switching, or add-on capture such as the FGD in Question 6) since $C_x$ scales linearly with $Q$; (3) site and time the release to favour good dispersion — avoid siting near terrain or building configurations that induce downwash (which defeats stack-height gains), and where operationally possible curtail or reduce emissions during forecast low-wind, stable (inversion) conditions when $\sigma_y,\sigma_z$ are smallest and $u$ is smallest, both of which independently drive $C_x$ up.