Gaussian plume dispersion of a power-plant SO₂ stack, and emission-reduction measures: 23-Chem-B2 Environmental Engineering, December 2014, 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	$H$	$40\ \text{m}$
SO₂ emission rate	$Q$	$20\ \text{g/min}=0.3333\ \text{g/s}$
Average wind speed	$u$	$12\ \text{m/s}$
Insolation	—	strong solar radiation
Stability class (Turner lookup)	—	$u>6\ \text{m/s}$, strong insolation ⇒ Class C
Class-C coefficients	$a,b,c,d,e,f$	$110,\ 1.0,\ -0.005,\ 60,\ 1.2,\ 0.02$
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 about 2 µg/m³.

Fig. 5: Predicted ground-level centerline SO₂ concentration vs. downwind distance for the 40 m stack, u=12 m/s (Class C). The curve peaks at ≈1.95 µg/m³ near x≈0.56 km, just under the 2 µg/m³ target — it never actually reaches the target line.

Approach. The Turner wind-speed/insolation lookup table selects the stability class before the dispersion coefficients can be evaluated: at 12 m/s (above the ≈6 m/s mechanical-turbulence threshold) with strong insolation, the table gives Class C, not the more-unstable A/B that "strong solar radiation" alone might otherwise suggest. Class-C 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.

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 C. With $x$ in kilometres, $$\sigma_y=110\,x^{\,1.0+0.005\ln x},\qquad \sigma_z=60\,x^{\,1.2-0.02\ln x}$$
Locate the peak (worst-case) concentration. A golden-section search on the numerical function locates the maximum at $$x_{peak}\approx0.56\ \text{km},\qquad C_{max}\approx1.95\ \mu\text{g/m}^3$$
Compare the peak to the 2 µg/m³ target. The peak ground-level concentration ($\approx1.95\ \mu\text{g/m}^3$) sits just below, but very close to, the 2 µg/m³ target — the curve rises toward the target line without ever crossing it. There is therefore no root $C(x)=2\ \mu\text{g/m}^3$ to solve for; the engineering conclusion is that the ground-level concentration stays "at about" (essentially just under) the 2 µg/m³ criterion at its single worst-case distance, and below it everywhere else. $$x\approx0.56\ \text{km — the point of closest approach to the target, still just under it}$$ ==**The ground-level SO₂ concentration peaks at ≈1.95 μg/m³ (x≈0.56 km) — it approaches but never quite crosses the 2 μg/m³ target at any downwind distance.**==

Quantity	Result
Peak ground-level concentration	≈1.95 µg/m³ at x≈0.56 km
Distance where C most nearly reaches 2 µg/m³	x ≈ 0.56 km (point of closest approach; target not crossed)

Check — sensitivity of the result to the stability-class assignment

Because the computed peak (1.95 µg/m³) sits so close to the 2 µg/m³ target, the conclusion is sensitive to the stability-class assignment: the standard Turner wind-speed/insolation table gives Class C for >6 m/s wind with strong insolation (used here), but if a Class D (neutral) assumption were instead applied the peak would be ≈5.5 µg/m³ — clearly above target, with a genuine crossing distance. The exam's own phrase "moderated unstable" is, in Turner's naming, Class B (moderately unstable); using the Class B row instead gives a peak of ≈5.1 µg/m³ at x≈0.26 km that falls back through 2 µg/m³ at x≈0.58 km — essentially the same distance as the Class C answer, so the reported x ≈ 0.56–0.58 km is robust to that reading. The Class-C result is retained as the correct table lookup for a 12 m/s wind with strong solar radiation, but the closeness of the margin is itself an engineering finding worth noting.

(ii) Engineering measures to reduce ground-level SO₂, compared by carbon footprint

Measure 1 — Flue-gas desulfurization (FGD). A wet limestone/lime scrubber downstream of the boiler removes SO₂ before it reaches the stack, directly cutting the emission rate $Q$ in the dispersion formula. Carbon footprint: the most effective source-reduction measure (routinely >90% SO₂ removal), but the scrubber's induced-draft fans, pumps and limestone-slurry preparation impose a parasitic auxiliary-power load on the plant, marginally raising its CO₂ intensity per unit net power delivered even though it does not touch the fuel-combustion CO₂ itself.

Measure 2 — Increase effective stack height. Raising $H$ increases the $\exp(-H^2/2\sigma_z^2)$ dilution term, lowering ground-level concentration at any given $x$. Carbon footprint: essentially carbon-neutral (a structural, one-time modification with no ongoing energy penalty) but it achieves no reduction in total SO₂ mass emitted — it only redistributes the same load over a larger area/greater distance, so it does nothing for the plant's actual carbon or SO₂ mass footprint, only the local ground-level peak.

Measure 3 — Fuel switching to lower-sulfur natural gas supply. Sourcing a pipeline gas blend with lower sulfur content reduces $Q$ at the source without any new equipment. Carbon footprint: genuinely reduces the SO₂ mass emitted with essentially no change in the plant's CO₂ intensity (natural gas combustion CO₂ is set by carbon content/heating value, largely independent of the trace sulfur content), making this the lowest-carbon-footprint of the three measures, though it is constrained by regional gas-supply sulfur specifications and may carry a fuel-price premium.

Preferred measure. Low-sulfur fuel switching is preferred on a carbon-footprint basis: it achieves genuine source-side SO₂ mass reduction (unlike stack-height increase) without FGD's parasitic auxiliary-power CO₂ penalty, making it the option with the smallest incremental carbon footprint for a comparable SO₂ benefit — though FGD remains the more robust choice where fuel-supply sulfur content cannot be controlled.

Concept & Theory

Topic. Gaussian plume ground-level concentration, Turner stability classification, and carbon-footprint comparison of SO₂ control strategies.

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 $Q$ but only through an exponential dilution term in $H$ and $\sigma_z$, which is why a 40 m stack combined with a brisk 12 m/s wind can keep even a real, continuous emission source right at the edge of a strict target; comparing control measures on carbon footprint (not just SO₂ removal) highlights that dispersion-only measures like stack height carry no CO₂ penalty but also no SO₂ mass benefit, whereas active treatment (FGD) trades a modest CO₂ penalty for a large SO₂ benefit.

Common pitfall. Assuming a root $C(x)=\text{target}$ must always exist and forcing a numerical solver to converge on a spurious crossing when the true peak lies only marginally below the target; treating "reduces SO₂" and "reduces carbon footprint" as the same axis of comparison when the question asks specifically for the latter.

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, compared by carbon footprint