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23-Chem-B2 Environmental Engineering · December 2015

Question 6 of 7: Gaussian plume dispersion of a coal-fired power-plant SO₂ stack, and emission-reduction measures

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. EGBC 04-Chem-B2 Environmental Engineering, December 2015, 3 hours, closed-book with a candidate-prepared double-sided 8½×11-inch aid sheet. Seven problems, each worth 20 marks; candidates attempt any five, and only the first five answers in the workbook are marked. All seven problems are solved below as a complete study resource.

Reference texts: G. Tchobanoglous, F. L. Burton & H. D. Stensel (Metcalf & Eddy), Wastewater Engineering: Treatment and Reuse (4th ed., McGraw-Hill) — BOD kinetics, dissolved air flotation, activated-sludge design; M. L. Davis & D. A. Cornwell, Introduction to Environmental Engineering (5th ed., McGraw-Hill) — drinking-water treatment, air pollution control, ion exchange, reverse osmosis, soil remediation; C. D. Cooper & F. C. Alley, Air Pollution Control: A Design Approach — cyclones, scrubbers, fabric filtration, electrostatic precipitation, odour control; S. P. Turner, Workbook of Atmospheric Dispersion Estimates (2nd ed., CRC Press) — the Gaussian plume model and Pasquill–Gifford stability classes. Canadian context follows the Canadian Environmental Protection Act (CEPA 1999), the Canadian Council of Ministers of the Environment (CCME) Municipal Wastewater Effluent and Drinking Water Quality guidelines, and provincial air/water permitting practice (e.g. BC Environmental Management Act, Metro Vancouver air-quality bylaws).

Question 6: Gaussian plume dispersion of a coal-fired 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.

QuantitySymbolValue
Stack (effective) height$H$$80\ \text{m}$
SO₂ emission rate$Q$$25\ \text{g/min}=0.4167\ \text{g/s}$
Average wind speed$u$$4\text{–}6\ \text{m/s}$ (range)
Sky condition—overcast
Stability class (Turner lookup)—overcast ⇒ neutral Class D at any wind speed
Class-D coefficients$a,b,c,d,e,f$$45,\ 0.8,\ -0.005,\ 60,\ 1.1,\ -0.06$
Target ground-level concentration$C$$5\ \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 5 µg/m³.

0.01.22.33.54.65.800.671.322.73.34downwind distance x (km)ground-level C (µg/m³)C = 5 µg/m³ (target)peak≈5.05stack H=80 m, Q=25 g/min, u=4 m/s (conservative bound), Class D
Fig. 5: Predicted ground-level centerline SO₂ concentration vs. downwind distance for the 80 m stack under the conservative low-wind bound (u=4 m/s, Class D). The curve marginally exceeds the 5 µg/m³ target (red markers) in a narrow window x≈0.95–1.09 km around the peak (≈5.05 µg/m³ at x≈1.01 km); at the average u=5 m/s the same stack stays comfortably under target everywhere (peak≈4.04 µg/m³).

Approach. Overcast sky conditions place the atmosphere in the neutral Turner stability Class D regardless of wind speed (unlike a clear-sky case, where the wind-speed/insolation table must be consulted), so the dispersion coefficients are fixed before evaluating $C(x)$. Because the problem specifies a wind-speed range (4–6 m/s) rather than a single value, and $u$ appears in the denominator of $C(x)$, the result is evaluated at the midpoint (5 m/s, the representative "average") for the primary answer, and separately at the conservative low-wind bound (4 m/s, which gives the highest ground-level concentration) to check whether the target is ever exceeded within the stated range.

  1. Ground-level centerline concentration formula. With a perfectly reflecting ground, $$C(x,0,0;H)=\frac{Q}{\pi\,u\,\sigma_y\sigma_z}\exp\!\left(-\frac{H^2}{2\sigma_z^2}\right)$$
  2. Dispersion coefficients for Class D. With $x$ in kilometres, $$\sigma_y=45\,x^{\,0.8+0.005\ln x},\qquad \sigma_z=60\,x^{\,1.1+0.06\ln x}$$
  3. Midpoint wind speed, u=5 m/s (primary "average" case). A numerical scan of $C(x)$ locates the peak at $$x_{peak}\approx1.01\ \text{km},\qquad C_{max}\approx4.04\ \mu\text{g/m}^3$$ which stays under the 5 µg/m³ target everywhere — at the average wind speed, the plant is already compliant at every downwind distance.
  4. Conservative low-wind bound, u=4 m/s. Repeating the peak search at the low end of the stated 4–6 m/s range (lower $u$ increases $C$, since $u$ sits in the denominator) gives $$x_{peak}\approx1.01\ \text{km},\qquad C_{max}\approx5.05\ \mu\text{g/m}^3$$ — here the peak marginally exceeds the 5 µg/m³ target (by only ≈1%). Bisecting on both sides of the peak locates the exceedance window: $$C(x)=5\ \mu\text{g/m}^3\ \text{at}\ x\approx0.946\ \text{km (rising)}\ \text{and}\ x\approx1.089\ \text{km (falling)}$$ so the concentration is above target only inside the narrow band $x\in[0.95,1.09]$ km, and permanently falls below target beyond $x\approx1.09$ km.
  5. Upper-wind bound, u=6 m/s (for completeness). The peak drops further to $C_{max}\approx3.37\ \mu\text{g/m}^3$, comfortably under target — the entire 4–6 m/s range therefore brackets a genuinely marginal case, with only the lowest 1 m/s of the range producing any exceedance at all.
QuantityResult
Peak concentration, u=5 m/s (average)≈4.04 µg/m³ at x≈1.01 km — never exceeds target
Peak concentration, u=4 m/s (conservative)≈5.05 µg/m³ at x≈1.01 km — marginally exceeds target
Distance beyond which C permanently falls below 5 µg/m³ (conservative case)x ≈ 1.09 km
Peak concentration, u=6 m/s≈3.37 µg/m³ — comfortably under target
Check — a genuinely marginal, wind-speed-sensitive case

Because the exam gives a wind-speed range rather than a single value, and this stack/emission combination happens to sit right at the 5 µg/m³ target, the compliance conclusion flips depending on which end of the 4–6 m/s range is used for design: compliant everywhere at the 5–6 m/s end, but marginally (≈1%) over target in a narrow ≈140 m-wide band near x≈1 km at the 4 m/s end. Standard conservative engineering practice for a compliance/design calculation is to use the wind speed that gives the worst-case (highest) ground-level concentration — here the low end of the range — so the defensible design answer to part (i) is x≈1.09 km (the point beyond which the target is permanently satisfied under the worst case within the stated range), while noting the exceedance is narrow and small in magnitude.

(ii) Engineering measures to reduce ground-level SO₂, prioritized by O&M requirement

Measure 1 (least O&M) — Increase effective stack height. Raising $H$ increases the $\exp(-H^2/2\sigma_z^2)$ dilution term, lowering ground-level concentration at every downwind distance and pushing the peak comfortably below target even at the conservative low-wind bound. Once built, a taller stack requires essentially no ongoing operating effort (periodic structural inspection only) — the lowest O&M measure of the three — but it achieves no reduction in total SO₂ mass emitted, only better ground-level dilution of the same load.

Measure 2 (moderate O&M) — Switch to a lower-sulfur coal supply. Sourcing coal with lower sulfur content directly reduces the emission rate $Q$ at the source. This genuinely reduces the SO₂ mass emitted (unlike Measure 1), with moderate ongoing effort — fuel-quality contracting/verification and potential boiler-performance adjustment for the different coal blend — but no new emissions-control hardware to operate.

Measure 3 (most O&M) — Flue-gas desulfurization (FGD). A wet limestone/lime scrubber downstream of the boiler removes SO₂ from the flue gas before it reaches the stack (routinely >90% removal), the most robust and largest reduction in emitted mass of the three measures, and effective regardless of weather/dispersion conditions. It carries the highest ongoing O&M burden: continuous limestone-slurry preparation and reagent supply, gypsum/wastewater by-product handling, and regular inspection/descaling of scrubber internals (mist eliminators, spray nozzles).

Recommended sequencing. Given this problem's exceedance is narrow and marginal (only the low end of the wind range, only ≈1% over target), stack-height increase is the natural first (lowest-O&M) measure to eliminate the exceedance; if a genuine mass-emission reduction is separately required (e.g. by a source-performance permit limit independent of dispersion), fuel switching offers a real reduction at moderate O&M before committing to the highest-O&M FGD option.