Atmospheric Dispersion of SO2/NOx: 23-Chem-B2 Environmental Engineering, May 2018, Q6

Problem 6: Atmospheric Dispersion of SO2/NOx (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.

Check: the exam names both SO2 (as the pollutant of concern) and NOx (as the quantity with a stated emission rate, 25 g/min from a 50 m stack); part (i) is worked treating the stated 25 g/min / 50 m release as the source term for the plume calculation (the pollutant label does not change the Gaussian dispersion mechanics). The 4000 GW power figure is not needed, since the emission rate is given directly. x is taken in km and σ in m, which gives σ values in the published Pasquill–Gifford range.

(i) Ground-level centerline concentration vs. downwind distance

Given.

Quantity	Value
Emission rate, Q	25 g/min = 0.417 g/s
Stack (effective release) height, H	50 m (no plume-rise data given)
Wind speed, u	5–10 m/s (design value u = 5 m/s, the worst case)
Insolation	Overcast sky
Target ground-level concentration	< 10 µg/m³

Find. The downwind distance x beyond which the predicted ground-level centerline concentration stays below 10 µg/m³.

Approach. Classify the Pasquill–Gifford stability class from wind speed + insolation, read a/b/c/d/e/f from the supplied table, build σy(x) and σz(x), then evaluate the ground-level (z=0) centerline (y=0) Gaussian-plume concentration $C(x,0,0,H)=\dfrac{Q}{\pi u\sigma_y\sigma_z}\exp\!\left[-\dfrac{1}{2}\left(\dfrac{H}{\sigma_z}\right)^2\right]$ over the downwind distance and check it against the 10 µg/m³ threshold.

Check: "overcast sky" is classified as neutral stability (Class D) for any wind speed, day or night — cloud cover suppresses the solar-heating-driven instability. The exam's generic phrase "moderated unstable dispersion parameters" is read as boilerplate; if an unstable Class C were used instead, the peak would be only 7.53 µg/m³ even at u = 5 m/s, so no exceedance would occur. Because C is proportional to 1/u, the low end of the stated range (u = 5 m/s) is the worst case and is used as the design value: a distance that holds at 5 m/s holds for every wind speed in the 5–10 m/s range.

Stability class. Overcast sky ⇒ neutral Class D. From the table: a=40, b=0.8, c=−0.005, d=80, e=1.3, f=−0.06.
Dispersion coefficients (log term in the exponent, x in km). $$\sigma_y=40\,x^{0.8+0.005\ln x},\qquad \sigma_z=80\,x^{1.3+0.06\ln x}$$ At x = 1 km both exponents reduce to b and e, so σy = 40 m and σz = 80 m; at x = 0.75 km, σy = 31.8 m and σz = 55.3 m.
Ground-level centerline concentration. Evaluating $C(x)=\dfrac{Q}{\pi u\sigma_y(x)\sigma_z(x)}\exp\!\left[-\dfrac12\left(\dfrac{H}{\sigma_z(x)}\right)^2\right]$ across x=0.01–5 km (numerically, since σy,σz are transcendental in x) traces the usual elevated-source shape: C(x) rises from 0 near the stack, peaks, then falls at larger x. See Fig. 3.
Locate the peak (u = 5 m/s). The maximum occurs at $$x_{max}\approx 0.57\ \text{km}, \qquad C_{max}\approx 11.75\ \mu\text{g/m}^3$$ which exceeds the 10 µg/m³ target, so a compliance distance exists. The concentration first rises above 10 µg/m³ at x ≈ 0.45 km.
Descending-limb crossing. Solving C(x) = 10 µg/m³ beyond the peak by bisection; check at x = 0.75 km: $$C=\frac{0.4167}{\pi(5)(31.8)(55.3)}\exp\!\left[-\frac12\left(\frac{50}{55.3}\right)^2\right]\times10^6=10.0\ \mu\text{g/m}^3$$ so $$\boxed{x \approx 0.75\ \text{km}\ (\approx 750\ \text{m})}$$ Beyond this distance the predicted ground-level SO2 concentration stays below 10 µg/m³. Sensitivity: at the mid-range wind (u = 7.5 m/s) the peak is only 7.83 µg/m³ and at 10 m/s it is 5.88 µg/m³, so for those winds the concentration never reaches the threshold. The 0.75 km distance is therefore the bound that holds across the whole stated wind range.

Fig. 3 — Predicted ground-level centerline SO2 concentration vs. downwind distance (Class D) at u = 5, 7.5 and 10 m/s, against the 10 µg/m³ threshold: only the worst-case 5 m/s curve exceeds it, falling back below 10 µg/m³ at x ≈ 0.75 km.

Quantity	Value
Stability class	D (neutral, overcast)
Design wind speed	5 m/s (low end of range, worst case)
Peak location, x_max	≈0.57 km
Peak ground-level concentration, C_max	≈11.75 µg/m³
σy, σz at the crossing	31.8 m, 55.3 m
Distance beyond which C < 10 µg/m³	≈0.75 km (750 m)

(ii) Engineering measures to reduce ground-level NOx, ranked least to most costly (25-year life-cycle)

Priority	Measure	25-year life-cycle cost basis	Long-term consideration
1 (least costly)	Combustion modification (low-NOx burners, staged combustion/flue-gas recirculation, excess-air/temperature tuning)	Moderate one-time capital (burner retrofit) plus routine tuning O&M; no major recurring reagent cost over the 25-year horizon.	Reduces NOx formation at the source (thermal-NOx suppression) rather than just redistributing the plume, but has a practical reduction ceiling (typically 30–50%) before more aggressive measures are needed.
2	Increase effective stack height / improve plume rise (taller stack or higher exit velocity/temperature)	Moderate-to-high capital (stack extension/rebuild), low incremental O&M; one-time capital amortized over 25 years with routine structural maintenance.	Lowers near-field ground-level concentration (roughly ∝1/H²) but does not cut total NOx mass emitted — redistributes and dilutes the same load farther downwind rather than reducing the regional NOx inventory.
3 (most costly)	Selective catalytic reduction (SCR) or selective non-catalytic reduction (SNCR) post-combustion NOx control	High capital (catalyst reactor/reagent-injection train) plus ongoing reagent (ammonia/urea) and catalyst-replacement O&M over the full 25-year life, with an annual catalyst-performance maintenance program.	Achieves the highest NOx reduction (SCR typically 80–90%+) and removes mass at the source rather than redistributing it, making it the only option of the three that meaningfully reduces the regional pollutant inventory.

Recommendation: combustion modification first (moderate cost, direct source reduction), stack-height increase only as an interim/complementary measure if near-field concentration remains a concern, and SCR/SNCR as the long-term solution for stringent NOx limits — consistent with the pollution-prevention hierarchy (reduce at source before diluting) over a 25-year equipment life.

Topic: Gaussian plume dispersion modelling and staged NOx/SO2 emission-reduction strategy.

Key relations: $C(x,0,0,H)=\dfrac{Q}{\pi u\sigma_y\sigma_z}\exp\!\left[-\dfrac12\left(\dfrac{H}{\sigma_z}\right)^2\right]$; Pasquill–Gifford σy(x), σz(x) power-law/ln correlations by stability class.

Why this works: The ground-level factor $\exp[-\frac12(H/\sigma_z)^2]$ is essentially zero right at the stack (σz small, H/σz huge) and only rises once σz has grown enough for the plume to reach the ground in appreciable concentration; it then falls again at large x purely because σyσz keep growing in the denominator faster than the exponential term keeps rising — hence the single interior maximum every elevated-source plume has. Whether that peak clears a given threshold depends on the interplay of stack height, emission rate, wind speed and stability class: here the 50 m, 25 g/min Class-D plume exceeds 10 µg/m³ only at the low end of the wind range, because C scales as 1/u.

Common pitfall: Assuming a stated wind-speed range or a qualitative sky description means the stability class can be skipped or guessed arbitrarily — "overcast" specifically fixes Class D independent of wind speed, unlike a clear-sky description where wind speed does shift the class; using the midpoint of a wind range, which can hide an exceedance at the lower wind speeds; flattening x^{b−c·ln x} into a·x^b−c·ln x (the two agree only at x = 1 km); also conflating "taller stack" with "less pollution" when it only redistributes the same mass over a larger area.

Source: Turner, Workbook of Atmospheric Dispersion Estimates, 2nd ed., Ch. 2–3 (Gaussian plume, Pasquill–Gifford stability classes); Cooper & Alley, Air Pollution Control, 4th ed., Ch. 3 (dispersion), Ch. 7 (NOx control technologies).

Question 6 of 7: Atmospheric Dispersion of SO2/NOx

Problem 6: Atmospheric Dispersion of SO2/NOx (20 marks)

(i) Ground-level centerline concentration vs. downwind distance

(ii) Engineering measures to reduce ground-level NOx, ranked least to most costly (25-year life-cycle)