Exercise set

Air Dispersion, Receptor Screening, and Emissions Inventory Exercises

Solved air-dispersion exercises for Gaussian plume screens, receptor margins, stack-height sensitivity, wind sensitivity and emissions inventory.

Branch: Environmental Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.0 · reviewed

These exercises focus on air dispersion screening and emissions inventory. They use simplified Gaussian plume, sensitivity and inventory calculations to estimate receptor scale, source contribution and reporting magnitude before deciding whether more detailed modeling or inventory review is required.

Assume simplified screening models unless an exercise states otherwise. Regulatory dispersion work may require approved meteorology, terrain, building downwash, plume rise, chemistry, deposition, background concentration and agency-specific averaging rules.

Release Evidence Notes

Dispersion evidence should state whether the calculation is a screen or an approved model result. A simple plume estimate can indicate scale, but it should not be reported as a final compliance model if the permit requires a specific modeling protocol.

Inventory evidence should identify throughput, emission factor, control efficiency, operating hours, material balance and uncertainty. Annual totals should not mix uncontrolled, controlled, startup and bypass modes without separate records.

Engineering Boundary Notes

These calculations do not replace approved dispersion modeling, emissions inventory certification, ambient monitoring, source testing or regulatory reporting. They are engineering screens for deciding whether a source, receptor or inventory claim needs escalation.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Plume screening	1, 2, 3, 4, 5, 6	Gaussian centerline, crosswind, stack height, wind and receptor margin	Decide whether a receptor screen is acceptable.
Source sensitivity	7, 8, 9, 10, 11	background, cumulative sources, building wake, deposition and odor screen	Decide whether detailed modeling is needed.
Emissions inventory	12, 13, 14, 15, 16, 17	annual mass, emission factor, control efficiency, uncertainty and evidence completion	Decide whether inventory records support reporting.
Release gate	18	all-of dispersion/inventory release	Decide whether the screening package can close.

Exercise 1: Gaussian Centerline Receptor Screen

A source emits $Q=0.20\ \text{g/s}$ . Wind speed is $u=4.0\ \text{m/s}$ , $\sigma_y=35\ \text{m}$ and $\sigma_z=18\ \text{m}$ . Use ground-level centerline screen:

C=\dfrac{Q}{\pi u\sigma_y\sigma_z}

Solution

C=\dfrac{0.20}{\pi(4.0)(35)(18)}

C=2.53\times10^{-5}\ \text{g/m}^3=25.3\ \mu\text{g/m}^3

Engineering Comment

This is a screening equation. Terrain, plume rise, downwash and averaging period can change the result.

Plausibility Check

A small gram-per-second source dispersed across thousands of square meters gives microgram-per-cubic-meter concentration.

Exercise 2: Receptor Margin

Modeled screening concentration is $25.3\ \mu\text{g/m}^3$ . Background is $18\ \mu\text{g/m}^3$ and screening criterion is $50\ \mu\text{g/m}^3$ . Compute margin.

Solution

Total:

C_t=25.3+18=43.3\ \mu\text{g/m}^3

Margin:

M=50-43.3=6.7\ \mu\text{g/m}^3

Engineering Comment

The screen passes with limited margin. Background basis and averaging period should be verified.

Plausibility Check

The source contribution plus background is below but close to the criterion.

Exercise 3: Crosswind Reduction

For a Gaussian plume, crosswind factor is:

F_y=\exp\left(-\dfrac{y^2}{2\sigma_y^2}\right)

Compute factor for $y=40\ \text{m}$ and $\sigma_y=35\ \text{m}$ .

Solution

F_y=\exp\left(-\dfrac{40^2}{2(35)^2}\right)

F_y=\exp(-0.653)=0.520

Engineering Comment

Crosswind offset can reduce concentration substantially, but wind direction variability may put receptors on centerline.

Plausibility Check

An offset a little larger than one sigma should reduce concentration to about half.

Exercise 4: Stack-Height Sensitivity

A screen estimates receptor concentration $40\ \mu\text{g/m}^3$ at effective stack height $20\ \text{m}$ . A height improvement is expected to reduce concentration by $30\%$ . Estimate new concentration.

Solution

C_2=40(1-0.30)=28\ \mu\text{g/m}^3

Engineering Comment

Stack height sensitivity should be checked with an approved model before using it for permitting.

Plausibility Check

A thirty percent reduction from forty gives twenty-eight.

Exercise 5: Wind-Speed Sensitivity

In a simple plume screen, concentration is inversely proportional to wind speed. If concentration is $36\ \mu\text{g/m}^3$ at $3\ \text{m/s}$ , estimate concentration at $6\ \text{m/s}$ .

Solution

C_2=C_1\dfrac{u_1}{u_2}=36\dfrac{3}{6}=18\ \mu\text{g/m}^3

Engineering Comment

Higher wind dilutes the plume in this simple screen, but stability and plume rise can alter the result.

Plausibility Check

Doubling wind speed halves concentration in the inverse model.

Exercise 6: Fence-Line Guard Band

Fence-line screen result is $42\ \mu\text{g/m}^3$ . The internal action level is $45\ \mu\text{g/m}^3$ with $5\ \mu\text{g/m}^3$ modeling guard. Decide status.

Solution

Guarded result:

C_g=42+5=47\ \mu\text{g/m}^3

Since:

47>45

the screen fails.

Engineering Comment

The nominal value is below the action level, but model uncertainty consumes the margin.

Plausibility Check

Only three micrograms per cubic meter nominal margin is less than the five-unit guard.

Exercise 7: Background Contribution Fraction

At a receptor, source contribution is $18\ \mu\text{g/m}^3$ and background is $27\ \mu\text{g/m}^3$ . Compute fraction from background.

Solution

Total:

C_t=18+27=45\ \mu\text{g/m}^3

Background fraction:

f=\dfrac{27}{45}=0.60=60\%

Engineering Comment

When background dominates, source reduction may not be enough to pass the receptor criterion.

Plausibility Check

Background is larger than source contribution and equals three fifths of total.

Exercise 8: Cumulative Source Screen

Three sources contribute $12$ , $9$ and $7\ \mu\text{g/m}^3$ at a receptor. Background is $14\ \mu\text{g/m}^3$ . Criterion is $45\ \mu\text{g/m}^3$ . Check margin.

Solution

C_t=12+9+7+14=42\ \mu\text{g/m}^3

M=45-42=3\ \mu\text{g/m}^3

Engineering Comment

The cumulative screen passes narrowly. A source-by-source permit decision should not ignore the combined condition.

Plausibility Check

The contributions sum just below the criterion.

Exercise 9: Building Wake Flag

A stack is $18\ \text{m}$ high near a building height of $14\ \text{m}$ . A screening rule flags downwash review when stack height is less than $1.5$ times building height. Check flag.

Solution

Threshold:

H_t=1.5(14)=21\ \text{m}

Since:

18<21

downwash review is flagged.

Engineering Comment

Building wake can increase ground-level concentrations and should not be ignored in the final model.

Plausibility Check

The stack is only modestly taller than the building, not one and a half times taller.

Exercise 10: Deposition Flux Screen

Air concentration is $30\ \mu\text{g/m}^3$ and deposition velocity is $0.006\ \text{m/s}$ . Estimate deposition flux.

Solution

F=Cv_d=30(0.006)=0.18\ \mu\text{g}/(\text{m}^2\text{s})

Engineering Comment

Deposition screens need pollutant form, surface type and averaging period before environmental significance can be judged.

Plausibility Check

A small deposition velocity multiplies concentration down to a small flux.

Exercise 11: Odor Dilution Screen

An odor source has measured concentration $900\ \text{OU/m}^3$ at the stack. A simple dilution estimate is $250:1$ at the receptor. Estimate receptor odor concentration.

Solution

C_r=\dfrac{900}{250}=3.6\ \text{OU/m}^3

Engineering Comment

Odor compliance often uses specific percentile and averaging rules. This is only a scale estimate.

Plausibility Check

Diluting by a few hundred reduces hundreds of odor units to a few odor units.

Exercise 12: Annual Emissions Inventory

A source emits $0.53\ \text{kg/h}$ for $2600\ \text{h/yr}$ . Compute annual emissions in tonnes per year.

Solution

M=0.53(2600)=1378\ \text{kg/yr}

M=1.378\ \text{t/yr}

Engineering Comment

Inventory records should separate normal, startup, shutdown and bypass hours if emission rates differ.

Plausibility Check

About half a kilogram per hour for a few thousand hours gives a little over one tonne per year.

Exercise 13: Emission Factor Inventory

A process uses $1800\ \text{t/yr}$ of material. Emission factor is $0.42\ \text{kg/t}$ before control. Control efficiency is $85\%$ . Compute controlled annual emissions.

Solution

Uncontrolled:

M_u=1800(0.42)=756\ \text{kg/yr}

Controlled:

M_c=756(1-0.85)=113.4\ \text{kg/yr}

Engineering Comment

Control efficiency must be supported by test or manufacturer evidence for the same pollutant and operating condition.

Plausibility Check

Eighty-five percent control leaves fifteen percent of the uncontrolled mass.

Exercise 14: Flare Destruction Inventory

A vent stream contains $22\ \text{kg/h}$ of VOC for $40\ \text{h}$ . Flare destruction efficiency is $98\%$ . Compute emitted VOC.

Solution

Inlet mass:

M_i=22(40)=880\ \text{kg}

Emitted:

M_e=880(1-0.98)=17.6\ \text{kg}

Engineering Comment

Flare efficiency depends on heating value, assist conditions and monitoring of flame presence.

Plausibility Check

Two percent of $880$ kg is under twenty kilograms.

Exercise 15: Inventory Uncertainty Guard

Reported annual emission is $8.4\ \text{t/yr}$ with uncertainty guard $1.1\ \text{t/yr}$ . Reporting threshold is $10\ \text{t/yr}$ . Decide status.

Solution

Guarded value:

M_g=8.4+1.1=9.5\ \text{t/yr}

Since:

9.5<10

the guarded inventory remains below threshold.

Engineering Comment

The margin is small. Throughput or factor updates could change reporting status.

Plausibility Check

Adding about one tonne to $8.4$ still leaves half a tonne of margin.

Exercise 16: Model Input Completeness

A dispersion screening package requires $8$ input records. Seven are accepted and meteorology is provisional. The gate requires all accepted. Decide status.

Solution

Accepted fraction:

C=\dfrac{7}{8}=87.5\%

Because meteorology is provisional and all records must be accepted, release is blocked.

Engineering Comment

Meteorology is not a secondary detail in dispersion work. Provisional meteorology should block final screening closure.

Plausibility Check

One provisional item fails a one hundred percent evidence gate.

Exercise 17: Inventory Evidence Completion

An inventory has throughput, emission factor and control efficiency accepted, but operating hours are estimated without source records. Decide whether final inventory evidence is complete.

Solution

The inventory equation needs:

M=Activity\times EF\times(1-\eta)

Operating hours are part of activity evidence. Since they are not source-record supported, final evidence is incomplete.

Engineering Comment

Inventory estimates can be useful internally, but reportable inventory needs traceable activity data.

Plausibility Check

Missing activity records affect the primary multiplier in the emissions estimate.

Exercise 18: Dispersion and Inventory Release Gate

A release gate requires receptor screen pass, downwash review pass, background basis pass, inventory evidence pass and uncertainty guard pass. Results are pass, fail, pass, pass and pass. Decide status.

Solution

The all-of gate fails because downwash review failed:

G=\text{blocked}

Engineering Comment

Receptor or inventory closure should not proceed when a required dispersion input review has failed.

Plausibility Check

Any failed condition blocks an all-of release gate.

Common Release Mistakes

Treating a Gaussian plume screen as an approved regulatory model.
Ignoring background concentration or cumulative sources.
Applying stack-height sensitivity without downwash review.
Mixing annual inventory modes with different emission rates.
Using emission factors without activity records.
Closing an inventory with provisional meteorology or incomplete control evidence.

Validation Package Checklist

Source emission rate, averaging period and receptor location.
Meteorology, background and terrain/downwash screening basis.
Gaussian plume or approved-model status clearly labeled.
Stack height, wind speed and sensitivity assumptions.
Annual throughput, operating hours, emission factor and control efficiency evidence.
Uncertainty guard and inventory threshold comparison.
Release gate with every dispersion and inventory input accepted.

REF