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23-Chem-B2 Environmental Engineering · May 2013

Question 5 of 7: Gaussian plume dispersion of a power-plant SO₂ stack

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

Notes on this paper

Paper format. EGBC 04-Chem-B2 Environmental Engineering, May 2013, 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, phosphorus removal; M. L. Davis & D. A. Cornwell, Introduction to Environmental Engineering (5th ed., McGraw-Hill) — air pollution control, ion exchange, reverse osmosis, soil remediation; L. Theodore & A. J. Buonicore / C. D. Cooper & F. C. Alley, Air Pollution Control: A Design Approach — fabric filtration, absorption, catalytic oxidation, 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 guidelines, and provincial air/water permitting practice (e.g. BC Environmental Management Act and Metro Vancouver air-quality bylaws), which govern effluent/emission limits, monitoring frequency, and buffer-strip / best-management-practice programs referenced throughout.

Question 5: Gaussian plume dispersion of a power-plant SO₂ stack (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$$40\ \text{m}$
SO₂ emission rate$Q$$30\ \text{g/min}=5\times10^5\ \mu\text{g/s}$
Average wind speed$u$$10\ \text{m/s}$
Stability—moderately unstable ⇒ Pasquill–Gifford Class B
Class-B coefficients$a,b,c,d,e,f$$100,\ 1.0,\ -0.004,\ 210,\ 1.0,\ 0.03$
Target ground-level concentration$C$$3\ \mu\text{g/m}^3$

Find. The downwind distance $x$ on the plume centerline ($y=0$) at ground level ($z=0$) where the predicted concentration falls to about 3 µg/m³.

03691200.10.20.30.40.50.60.7downwind distance x (km)ground-level C (µg/m³)C = 3 µg/m³ (target)x≈0.49 kmpeak ≈13.6stack H=40 m, Q=30 g/min, u=10 m/s, Class B
Fig. 5: Predicted ground-level centerline SO₂ concentration vs. downwind distance for the 40 m stack (Class B). The concentration is zero at the stack, rises sharply as the plume touches the ground, peaks at ≈13.6 µg/m³ near x≈0.15 km, then falls monotonically; it re-crosses the 3 µg/m³ target at x≈0.49 km.

Approach. “Moderate solar radiation” identifies the atmosphere as moderately unstable, i.e. Pasquill–Gifford Class B, so the Class-B row of the coefficient table gives $\sigma_y(x)$ and $\sigma_z(x)$ directly. These feed the standard ground-level, on-axis Gaussian plume formula (which already includes the ground-reflection image term), and because $\sigma_y,\sigma_z$ are themselves power-law functions of $x$, the resulting concentration-vs-distance relation must be solved for $x$ numerically/graphically rather than algebraically.

  1. Ground-level centerline concentration formula. With a perfectly reflecting ground (no deposition), the plume equation reduces on the centerline ($y=0$) at ground level ($z=0$) to $$C(x,0,0;H)=\frac{Q}{\pi\,u\,\sigma_y\sigma_z}\exp\!\left(-\frac{H^2}{2\sigma_z^2}\right)$$ where the factor of 2 from the ground-reflected image source cancels the usual $2\pi$ denominator, leaving $\pi$.
  2. Dispersion coefficients for Class B. With $x$ in kilometres, $$\sigma_y=100\,x^{\,1.0+0.004\ln x},\qquad \sigma_z=210\,x^{\,1.0-0.03\ln x}$$ (using $b-c\ln x=1.0-(-0.004)\ln x$).
  3. Evaluate $C(x)$ over a range of $x$ and locate the target. Substituting the Step 2 expressions into Step 1 and scanning $x$ shows $C$ rising steeply from zero at the stack, peaking at $$C_{max}\approx13.6\ \mu\text{g/m}^3\ \text{near}\ x\approx0.15\ \text{km}$$ (the point where the growing $\sigma_y\sigma_z$ in the denominator first overtakes the shrinking exponential penalty for the plume not yet having spread down to the ground), then falling monotonically as the plume continues to spread. The target 3 µg/m³ is therefore crossed twice; the question's wording — the concentration “falls to” 3 µg/m³ — identifies the far-side (post-peak, decreasing) crossing as the one asked for.
  4. Solve for $x$ on the decreasing branch (numerical root). Bisecting $C(x)=3\ \mu\text{g/m}^3$ on the branch beyond the peak gives $$x\approx0.486\ \text{km}\approx490\ \text{m}$$ ==**Downwind distance x ≈ 0.49 km (≈490 m) from the stack**==
QuantityResult
$\sigma_y,\sigma_z$ at $x=0.49$ km$48.7\ \text{m},\ 100.6\ \text{m}$ (approx.)
Peak ground-level concentration$C_{max}\approx13.6\ \mu\text{g/m}^3$ at $x\approx0.15$ km
Distance where $C$ falls to 3 µg/m³x ≈ 0.49 km (≈490 m)
Check — stability class and effective stack height

“Moderately unstable” is read as Pasquill–Gifford Class B directly from the standard nomenclature (A=extremely unstable, B=moderately unstable, C=slightly unstable, D=neutral); note that the classical Pasquill wind-speed/insolation lookup table would actually assign Class D at $u=10$ m/s regardless of daytime insolation (wind speeds >6 m/s default to neutral), so the question's explicit “moderately unstable” instruction is taken as overriding that lookup and directly selecting the Class-B row. Sensitivity: the same equations put the 3 µg/m³ falling-limb distance at ≈0.70 km with the Class C row and ≈1.28 km with the Class D row, so state the class assumption explicitly on the exam. $H=40$ m is taken as the effective stack height (physical height, no plume-rise correction given/needed).

(ii) Engineering measures to reduce ground-level SO₂ concentration

Measure 1 — increase the effective stack height. Raising the physical stack (or adding a hot, high-exit-velocity plume that gains buoyant/momentum plume rise) increases $H$ in the exponential term above, which sharply lowers the ground-level concentration near the plant (the near-field peak scales roughly as $1/H^2$) by delaying where the plume first touches the ground. Environmental impact: this reduces local, near-source ground-level concentration but does not reduce total SO₂ mass emitted — it simply disperses the same load over a larger area and greater downwind distance, which historically contributed to long-range transboundary acid rain (a documented Canada–U.S. transboundary air issue). It is therefore a locally effective but not an environmentally preferred long-term solution.

Measure 2 — flue-gas desulfurization (wet limestone scrubbing). Installing an absorption-based FGD system removes SO₂ from the flue gas before it ever reaches the stack, directly cutting $Q$ in the plume equation (concentration scales linearly with $Q$) rather than just redistributing it. Environmental impact: this is a true source-reduction measure (typically 90–98% SO₂ removal) with the added benefit of a saleable/landfillable gypsum by-product, but it has a real capital/operating cost and its own waste stream (scrubber sludge/wastewater) to manage.

Measure 3 — fuel switching / desulfurized fuel. Switching to a lower-sulfur coal, or to natural gas, reduces the SO₂ generated at the source in the first place (again lowering $Q$ directly). Environmental impact: also a genuine source-reduction measure, with the co-benefit of reduced particulate and mercury emissions if switching away from coal entirely, but it may raise fuel cost and, for a switch to natural gas, shifts (does not eliminate) the facility's overall greenhouse-gas profile depending on the displaced fuel's carbon intensity.

Recommendation. Flue-gas desulfurization is the preferred option: unlike stack-height increase (Measure 1), it achieves a genuine reduction in total SO₂ mass released rather than exporting the problem downwind, and unlike a full fuel switch (Measure 3) it can be retrofitted onto the existing 300 GW plant without abandoning existing fuel-supply infrastructure; a combination of FGD plus a modest stack-height increase, as is common Canadian utility practice, addresses both the near-field ground-level concentration and the total emitted load simultaneously.