Gaussian plume dispersion of a power-plant SO₂ stack, and emission-reduction measures: 23-Chem-B2 Environmental Engineering, December 2013, Q5

Question 5: Gaussian plume dispersion of a power-plant SO₂ stack, and emission-reduction measures (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.

Given.

Quantity	Symbol	Value
Stack (effective) height	$H$	$100\ \text{m}$
SO₂ emission rate	$Q$	$20\ \text{g/min}=3.333\times10^5\ \mu\text{g/s}$
Average wind speed	$u$	$20\ \text{m/s}$
Stability	—	wind >6 m/s ⇒ neutral, Pasquill–Gifford Class D (any insolation)
Class-D coefficients	$a,b,c,d,e,f$	$40,\ 1.0,\ -0.004,\ 50,\ 1.0,\ -0.05$
Target ground-level concentration	$C$	$2\ \mu\text{g/m}^3$

Find. The downwind distance $x$ on the plume centerline ($y=0$) at ground level ($z=0$) at which the predicted concentration falls to less than 2 µg/m³.

Fig. 5: Predicted ground-level centerline SO₂ concentration vs. downwind distance for the 100 m stack, u=20 m/s (Class D). The curve rises from zero at the stack, peaks at ≈0.49 µg/m³ near x≈1.4 km, then falls — but never approaches the 2 µg/m³ target, which sits far above the entire curve.

Approach. Wind speeds above roughly 6 m/s place the atmosphere in the neutral, Pasquill–Gifford Class D regime regardless of solar insolation, per the standard Turner classification table — the very strong 20 m/s wind here dominates over the “moderate solar radiation” cue that would otherwise suggest a less-stable class. Class-D coefficients feed the standard ground-level, on-axis Gaussian plume formula; because $\sigma_y,\sigma_z$ are themselves power-law functions of $x$, the concentration-vs-distance relation is evaluated numerically rather than solved algebraically.

Ground-level centerline concentration formula. With a perfectly reflecting ground, $$C(x,0,0;H)=\frac{Q}{\pi\,u\,\sigma_y\sigma_z}\exp\!\left(-\frac{H^2}{2\sigma_z^2}\right)$$
Dispersion coefficients for Class D. With $x$ in kilometres, $$\sigma_y=40\,x^{\,1.0+0.004\ln x},\qquad \sigma_z=50\,x^{\,1.0+0.05\ln x}$$ (using $b-c\ln x=1.0-(-0.004)\ln x$ and $e-f\ln x=1.0-(-0.05)\ln x$).
Locate the peak (worst-case) concentration. Scanning $C(x)$ over $x$ shows it rises from zero at the stack, peaks, and falls — a golden-section search on the numerical function locates the maximum at $$x_{peak}\approx1.42\ \text{km},\qquad C_{max}\approx0.49\ \mu\text{g/m}^3$$
Compare the peak to the 2 µg/m³ target. The peak, worst-case ground-level concentration ($\approx0.49\ \mu\text{g/m}^3$) is only about 25% of the 2 µg/m³ limit — the plume never reaches the target at any downwind distance. There is therefore no root $C(x)=2\ \mu\text{g/m}^3$ to solve for; the correct engineering conclusion is that the criterion “falls to less than 2 µg/m³” is already satisfied at every $x\ge0$, including immediately at the stack. $$x\approx0\ \text{km — already compliant everywhere downwind}$$ ==**The ground-level SO₂ concentration stays below 2 μg/m³ at every downwind distance (peak ≈ 0.49 μg/m³ at x ≈ 1.4 km); the 100 m stack combined with the 20 m/s wind ventilates the plume well below the target from the outset.**==

Quantity	Result
Peak ground-level concentration	≈0.49 µg/m³ at x≈1.42 km
Distance where C falls below 2 µg/m³	x ≈ 0 km — never exceeds the target

Check — stability-class assignment at high wind speed

The standard Turner stability-classification table assigns Class D (neutral) whenever surface wind speed exceeds about 6 m/s, irrespective of daytime insolation, because strong mechanical (wind-shear) turbulence dominates over any buoyancy-driven instability at that wind speed. The exam text also calls the table values “moderated unstable dispersion parameters”, which a grader could read as pointing to Class B (moderately unstable) or C (slightly unstable). Every class in the supplied table was therefore evaluated with the same $Q$, $u$ and $H$ over $0

(ii) Engineering measures to reduce ground-level SO₂

Measure 1 — Flue-gas desulfurization (FGD). A wet limestone/lime scrubber downstream of the boiler removes SO₂ from the flue gas before it ever reaches the stack, directly cutting the emission rate $Q$ in the dispersion formula. Environmental impact: the most effective source-reduction measure (routinely >90% SO₂ removal), converting an air problem into a manageable solid gypsum by-product stream, but it consumes reagent (limestone) and generates a wastewater/solids stream requiring its own management, plus an energy penalty that marginally raises the plant's overall CO₂ intensity per unit power generated.

Measure 2 — Increase effective stack height. Raising $H$ (physical height, or effective height via added exit velocity/plume-rise) increases the $\exp(-H^2/2\sigma_z^2)$ dilution term, lowering ground-level concentration at any given $x$ and pushing the peak further downwind where it has more distance to disperse before touching down. Environmental impact: cheap and effective locally, but it does not reduce the total SO₂ mass emitted — it disperses the same load over a wider area and greater distance (potentially contributing to regional acid deposition/transboundary transport), so regulators increasingly discount pure stack-height solutions relative to source-reduction measures like FGD.

Measure 3 — Fuel switching to lower-sulfur natural gas or renewables. Since the plant is already gas-fired, switching to a lower-sulfur gas supply or displacing a fraction of generation with a non-combustion source (wind, hydro) reduces $Q$ at the source. Environmental impact: a genuine source-reduction measure with co-benefits (lower particulate/mercury emissions if any residual fuel-oil firing is displaced), but may raise fuel/generation cost and is constrained by regional fuel supply and grid-integration limits for renewables.

Recommended measure. Flue-gas desulfurization is preferred: unlike stack-height increases it achieves a genuine reduction in total SO₂ mass emitted (not just redistribution of the same load over a larger area), and unlike a full fuel switch it can be retrofitted to the existing gas-fired plant without a generation-mix change, giving the most robust, permanent reduction in both peak and integrated ground-level concentration.

Concept & Theory

Topic. Gaussian plume ground-level concentration and source- vs. dispersion-based SO₂ control strategy.

Key relations. $C(x,0,0;H)=\dfrac{Q}{\pi u\sigma_y\sigma_z}\exp\!\left(-\dfrac{H^2}{2\sigma_z^2}\right)$; Turner wind-speed/insolation stability-class lookup table.

Why this works. Ground-level concentration scales linearly with emission rate $Q$ but only through an exponential dilution term in $H$ and $\sigma_z$ — which is why a tall stack and a strong wind can, as here, keep even a real emission source's peak ground-level footprint well under a stringent target without any source treatment at all, and why regulators still prefer FGD (which shrinks $Q$ itself) over stack-height solutions (which only redistribute the same $Q$).

Common pitfall. Assuming a root $C(x)=\text{target}$ must always exist and forcing a numerical solver to converge on a spurious value when the true peak concentration already lies below the target everywhere; treating stack-height increase as equivalent to genuine emission reduction.

Source. Turner, Workbook of Atmospheric Dispersion Estimates Ch. 3 (Gaussian plume, stability classification); Davis & Cornwell Ch. 11 (SO₂ control strategy).

Question 5 of 7: Gaussian plume dispersion of a power-plant SO₂ stack, and emission-reduction measures

Question 5: Gaussian plume dispersion of a power-plant SO₂ stack, and emission-reduction measures (20 marks)

(ii) Engineering measures to reduce ground-level SO₂