Cyclone Collection Efficiency and Emissions Trading: 18-Env-A5 Air Quality and Pollution Control Engineering, December 2019, Q6

Problem 6: Cyclone Collection Efficiency and Emissions Trading (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.

Part (i) — overall cyclone collection efficiency.

Given. Cut diameter $d_{pc}=6\ \mu\text{m}$; four particle-size ranges with their mass-fraction distribution.

Size range $d_{pj}$ (µm)	Mass fraction $m_j/M$
1–10	10%
10–20	30%
20–50	40%
50–100	20%

Find. The overall (mass-weighted) collection efficiency of the cyclone.

Approach. Represent each size range by its arithmetic midpoint diameter, compute the range's individual removal efficiency $\eta_j$ from the supplied formula, then combine the four ranges by their mass fraction to get the overall efficiency $\eta_{overall}=\sum(\eta_j\cdot m_j/M)$.

Midpoint diameters. $\bar d_{p,1}=5.5$, $\bar d_{p,2}=15$, $\bar d_{p,3}=35$, $\bar d_{p,4}=75\ \mu\text{m}$ (the arithmetic mean of each printed range, a standard simplifying representation — see the check note).
Individual range efficiencies. $\eta_j=1/(1+(d_{pc}/\bar d_{p,j})^2)$: $$\eta_1=\frac{1}{1+(6/5.5)^2}=0.457,\quad\eta_2=\frac{1}{1+(6/15)^2}=0.862,\quad\eta_3=\frac{1}{1+(6/35)^2}=0.971,\quad\eta_4=\frac{1}{1+(6/75)^2}=0.994$$
Mass-weighted overall efficiency. $$\eta_{overall}=\sum \eta_j\left(\frac{m_j}{M}\right)=0.457(0.10)+0.862(0.30)+0.971(0.40)+0.994(0.20)=\boxed{89.2\%}$$

Size range (µm)	$\eta_j$	Weighted contribution
1–10	45.7%	4.57%
10–20	86.2%	25.86%
20–50	97.1%	38.86%
50–100	99.4%	19.87%
Overall efficiency		89.2%

Check: assumption — each printed size range is represented by its arithmetic midpoint diameter (5.5, 15, 35, 75 µm), the standard simplification for a mass-fraction/grade-efficiency table when a full particle-size distribution curve is not supplied (Cooper & Alley). Because $\eta_j$ is a concave, rapidly saturating function of $d_{pj}$ above $d_{pc}$, this slightly understates the true efficiency of the two coarser bins relative to a log-mean or distribution-weighted diameter, but the effect on the overall 89.2% figure is at most a percentage point or two.

Part (ii) — emissions trading. Under a cap-and-trade emissions-trading scheme, a regulator sets an aggregate emissions cap for a pollutant across a sector or region (typically declining over successive compliance periods) and issues tradeable allowances (credits) totaling that cap, either by free allocation or auction. A source whose actual emissions fall below its allocated allowances may sell the surplus; a source facing a higher marginal abatement cost may instead purchase allowances to cover its shortfall rather than installing additional controls. Because trading lets abatement occur wherever it is cheapest across the whole market, the same aggregate emissions cap is achieved at a lower total (society-wide) cost than a uniform technology or emission-rate standard applied identically to every source. Two disadvantages compared with direct source controls: (1) localized "hot spots" — a source can meet its compliance obligation entirely by purchasing credits rather than reducing its own stack emissions, so ambient concentrations near that specific source (and its exposed neighbours) can remain high even though the regional aggregate total is within the cap; (2) weaker technology-forcing and administrative complexity — trading does not guarantee that any individual source actually installs cleaner technology (it only guarantees the aggregate total), and the scheme itself requires robust emissions monitoring/verification and is exposed to allowance-price volatility and market-design gaming, adding regulatory and compliance-planning complexity that a fixed emission-rate standard does not have.

Topic: Grade-efficiency-weighted cyclone performance; cap-and-trade emissions-trading policy design.

Key relations: cyclone grade-efficiency curve $\eta_j=1/(1+(d_{pc}/d_{pj})^2)$; overall (mass-weighted) efficiency $\eta_{overall}=\sum\eta_j(m_j/M)$; cap-and-trade aggregate cap with tradeable allowances equalizing marginal abatement cost across sources.

Why this works: a cyclone's efficiency is strongly size-dependent, so the correct "overall" figure for a polydisperse dust must be a mass-weighted average across the size distribution, not the efficiency evaluated at a single representative particle size; emissions trading works because the environmental benefit of an aggregate cap does not depend on which source achieves the required reduction, only that the total is met, so letting the market find the cheapest combination of reductions minimizes total abatement cost.

Common pitfall: evaluating $\eta_j$ once at the overall mean particle size instead of weighting each size range's own efficiency by its mass fraction (this understates efficiency for a distribution skewed toward coarse particles, as here); presenting emissions trading as strictly superior to source controls without noting the hot-spot equity concern.

Source: Cooper & Alley, Air Pollution Control: A Design Approach, 4th ed., Ch. 5 (cyclone design and grade efficiency); Environment and Climate Change Canada, CEPA 1999 cap-and-trade policy guidance.

Question 6 of 7: Cyclone Collection Efficiency and Emissions Trading

Problem 6: Cyclone Collection Efficiency and Emissions Trading (20 marks)