Exercise set

Air Emissions Stack Monitoring, CEMS, and Compliance Evidence Exercises

Solved air-emissions exercises for stack mass rate, gas corrections, analyzer drift, CEMS valid hours, rolling averages and compliance release.

Branch: Environmental Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

These exercises focus on stack monitoring and compliance evidence. They convert measured flow and concentration into mass rate, correct gas data to the required basis, check analyzer validity, evaluate CEMS completeness and decide whether a compliance claim is defensible.

Assume simplified screening calculations unless an exercise states otherwise. Actual reporting must follow the permit, approved test method, quality-assurance plan, missing-data rule, averaging period and regulatory basis.

How to use these exercises

Use the set as a compliance-evidence review. Exercises 1 to 6 establish the reporting basis: mass rate, guarded permit margin, dry flow, oxygen correction, pollutant load and valid stack-test averages. Exercises 7 to 14 test analyzer and CEMS validity through SNR, drift, detection limit, valid hours, rolling averages, missing-data substitution and calibration gas tolerance. Exercises 15 to 18 connect the measurements to control efficiency, exceedance duration, corrective-action closure and release status.

For every calculation, record the permit condition, pollutant, averaging period, gas basis and data-validity rule. A measured value is not compliance evidence until the basis and validity rule match the permit.

Release Evidence Notes

Stack evidence must preserve basis and averaging time. Flow can be actual, standard, wet, dry or oxygen-corrected. Concentration can be raw, moisture-corrected, oxygen-corrected or drift-adjusted. The release package should state the exact basis before comparing with a limit.

Monitoring evidence should distinguish measured from valid. Analyzer drift, calibration status, detection limit, signal quality, valid-hour rules and event records determine whether a reported average is usable.

The package should also separate operation release from event closure. A source may be technically back under a limit while the compliance record is still blocked by missing maintenance evidence, invalid-hour substitution, calibration documentation, notification requirements or corrective-action sign-off.

Engineering Boundary Notes

These exercises are screening calculations. They do not replace approved stack testing, CEMS certification, relative accuracy testing, regulatory missing-data substitution or permit-specific compliance procedures.

Real compliance work depends on the approved method, reference conditions, sample location, traverse point validity, moisture and oxygen measurements, span range, calibration gas traceability, data acquisition system flags, downtime records, bypass definitions, event notification rules and regulator-approved QA plans. Use these exercises to find evidence mismatches before comparing a number with a limit.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Stack reporting basis	1, 2, 3, 4, 5, 6	mass rate, dry correction, oxygen correction, averaging and guard band	Decide whether stack values match the permit basis.
Analyzer and CEMS validity	7, 8, 9, 10, 11, 13, 14	SNR, drift, detection limit, valid hours, rolling average and calibration gas	Decide whether monitoring data are valid.
Compliance closure	12, 15, 16, 17, 18	bypass mass, control efficiency, exceedance duration, closure score and release gate	Decide whether operation or event closure is defensible.

Exercise 1: Stack Emission Mass Rate

A stack exhausts $Q=3.2\ \text{m}^3/\text{s}$ with pollutant concentration $C=46\ \text{mg/m}^3$ . Compute mass rate in $\text{g/s}$ .

Solution

\dot m=QC=3.2(46)=147.2\ \text{mg/s}

\dot m=0.147\ \text{g/s}

Engineering Comment

The result is valid only on the stated flow and concentration basis. Wet/dry or standard/actual mismatch can dominate the error.

Plausibility Check

A few cubic meters per second at tens of milligrams per cubic meter gives tenths of grams per second.

Exercise 2: Permit Mass-Rate Margin

The permit limit is $0.62\ \text{kg/h}$ . A test reports $0.53\ \text{kg/h}$ and uncertainty guard $0.04\ \text{kg/h}$ . Check compliance margin.

Solution

Guarded result:

E_g=0.53+0.04=0.57\ \text{kg/h}

Margin:

M=0.62-0.57=0.05\ \text{kg/h}

Engineering Comment

The source passes the guarded screen, but with limited margin. Operating conditions should match the permitted mode.

Plausibility Check

The guarded result remains below the limit by five hundredths of a kilogram per hour.

Exercise 3: Dry Standard Flow Correction

Measured wet flow is $12000\ \text{Nm}^3/\text{h}$ with moisture fraction $0.12$ . Compute dry standard flow.

Solution

Q_{dry}=Q_{wet}(1-B_w)

Q_{dry}=12000(1-0.12)=10560\ \text{Nm}^3/\text{h}

Engineering Comment

Dry correction must use the moisture basis required by the permit or method.

Plausibility Check

Removing twelve percent moisture should reduce flow by twelve percent.

Exercise 4: Oxygen-Corrected Concentration

Measured concentration is $85\ \text{mg/Nm}^3$ at $9\%$ oxygen. Correct to $6\%$ oxygen using:

C_{corr}=C_{meas}\dfrac{21-O_{ref}}{21-O_{meas}}

Solution

C_{corr}=85\dfrac{21-6}{21-9}=85\dfrac{15}{12}=106.25\ \text{mg/Nm}^3

Engineering Comment

Oxygen correction penalizes dilution. The oxygen measurement must be valid over the same averaging period.

Plausibility Check

Correcting from nine percent oxygen to six percent increases the concentration.

Exercise 5: Pollutant Load Over an Averaging Period

An emission rate is $0.42\ \text{kg/h}$ for $6$ hours. Compute pollutant load.

Solution

M=0.42(6)=2.52\ \text{kg}

Engineering Comment

Load evidence should state whether the rate was constant, averaged or integrated from valid CEMS data.

Plausibility Check

About half a kilogram per hour for several hours gives a few kilograms.

Exercise 6: Guarded Stack-Test Average

Three valid test runs report $41$ , $44$ and $43\ \text{mg/Nm}^3$ . The limit is $50\ \text{mg/Nm}^3$ and guard is $3\ \text{mg/Nm}^3$ . Decide status.

Solution

Average:

\bar C=\dfrac{41+44+43}{3}=42.7\ \text{mg/Nm}^3

Guarded result:

C_g=42.7+3=45.7\ \text{mg/Nm}^3

Since $45.7<50$ , the test passes.

Engineering Comment

Every run must be valid and representative. Averaging invalid runs is not compliance evidence.

Plausibility Check

All runs are in the low forties, so the guarded average remains below fifty.

Exercise 7: Analyzer Signal-to-Noise Ratio

An analyzer signal is $2.4\ \text{V}$ and RMS noise is $0.08\ \text{V}$ . Compute SNR.

Solution

SNR=\dfrac{2.4}{0.08}=30

Engineering Comment

SNR should be checked near the relevant concentration range, not only at high span.

Plausibility Check

Noise is about one thirtieth of signal.

Exercise 8: Calibration Drift Correction

An analyzer reads $102\ \text{ppm}$ on a $100\ \text{ppm}$ calibration gas after the run. Estimate drift percentage.

Solution

d=\dfrac{102-100}{100}=0.02=2\%

Engineering Comment

Whether this passes depends on the method drift criterion and whether correction is allowed.

Plausibility Check

A two ppm error on a one hundred ppm gas is two percent.

Exercise 9: Detection-Limit Release Screen

A permit limit is $5.0\ \text{ppm}$ . The method detection limit is $1.4\ \text{ppm}$ . The QA plan requires detection limit below $25\%$ of the limit. Check status.

Solution

Allowed detection limit:

DL_{max}=0.25(5.0)=1.25\ \text{ppm}

Since:

1.4>1.25

the method fails the screen.

Engineering Comment

The method may not be sensitive enough to support compliance near the limit.

Plausibility Check

The detection limit is slightly above one quarter of the limit.

Exercise 10: CEMS Valid Hours

A month has $720$ hours. CEMS data are valid for $692$ hours. The permit requires $95\%$ valid hours. Check status.

Solution

V=\dfrac{692}{720}=0.961=96.1\%

Since $96.1\%>95\%$ , the month passes.

Engineering Comment

Valid-hour percentage does not excuse missing data during high-emission events if specific rules apply.

Plausibility Check

Only twenty-eight hours are invalid, less than five percent of the month.

Exercise 11: Rolling-Average Compliance

Four hourly averages are $42$ , $47$ , $50$ and $45\ \text{ppm}$ . The rolling four-hour limit is $48\ \text{ppm}$ . Check status.

Solution

\bar C=\dfrac{42+47+50+45}{4}=46\ \text{ppm}

Since:

46<48

the rolling average passes.

Engineering Comment

The hour at $50$ ppm may still matter if an hourly cap or excursion rule exists.

Plausibility Check

The average of values around the mid-forties is below forty-eight.

Exercise 12: Bypass Event Emission Mass

A bypass emits $0.18\ \text{kg/h}$ for $42\ \text{min}$ . Compute emitted mass.

Solution

Time:

t=\dfrac{42}{60}=0.70\ \text{h}

Mass:

M=0.18(0.70)=0.126\ \text{kg}

Engineering Comment

Bypass closure should include cause, duration, control status, notification rule and corrective action.

Plausibility Check

Less than one hour at less than a quarter kilogram per hour gives about one tenth of a kilogram.

Exercise 13: Missing-Data Substitution

A rule requires substituting $95$ th percentile value $72\ \text{ppm}$ for $3$ invalid hours. Valid hours around the event average $44\ \text{ppm}$ for $21$ hours. Compute substituted 24-hour average.

Solution

\bar C=\dfrac{21(44)+3(72)}{24}

\bar C=\dfrac{924+216}{24}=47.5\ \text{ppm}

Engineering Comment

Missing data can materially affect compliance averages. The substitution rule must be applied exactly.

Plausibility Check

Three high substitute hours pull the average above the valid-hour average.

Exercise 14: Calibration Gas Tolerance

A span gas certificate says $100.0\ \text{ppm}$ with tolerance $\pm1.0\%$ . Compute acceptable concentration range.

Solution

Tolerance:

T=0.01(100.0)=1.0\ \text{ppm}

Range:

99.0\le C\le101.0\ \text{ppm}

Engineering Comment

Calibration evidence should include certificate, expiration and traceability.

Plausibility Check

One percent of one hundred ppm is one ppm.

Exercise 15: Control Device Efficiency

Inlet pollutant mass rate is $12.5\ \text{kg/h}$ and outlet is $0.75\ \text{kg/h}$ . Compute control efficiency.

Solution

\eta=1-\dfrac{0.75}{12.5}=1-0.06=0.94

\eta=94\%

Engineering Comment

Efficiency should be paired with outlet compliance. High removal can still fail if inlet loading is high.

Plausibility Check

Outlet is six percent of inlet, so removal is ninety-four percent.

Exercise 16: Exceedance Duration

A concentration exceeds a permit threshold for $7$ consecutive $15$ minute data blocks. Compute exceedance duration.

Solution

t=7(15)=105\ \text{min}=1.75\ \text{h}

Engineering Comment

Duration determines reporting and corrective-action requirements. Timestamp integrity matters.

Plausibility Check

Four blocks are one hour; seven blocks are one and three quarter hours.

Exercise 17: Corrective-Action Closure Score

A closure checklist has $10$ required records. Nine are accepted and one maintenance record is missing. Gate requires all accepted. Decide status.

Solution

Accepted percentage:

C=\dfrac{9}{10}=90\%

Because one required record is missing, closure is blocked.

Engineering Comment

Compliance closure needs records, not only improved readings after the event.

Plausibility Check

Any missing record fails an all-required closure rule.

Exercise 18: Stack Monitoring Compliance Release Gate

A release gate requires valid stack basis, analyzer QA pass, CEMS completeness pass, rolling average pass and corrective-action closure pass. Results are pass, pass, pass, pass and fail. Decide status.

Solution

The all-of gate fails because corrective-action closure failed:

G=\text{blocked}

Engineering Comment

Normal operation should not be released if event closure evidence remains incomplete.

Plausibility Check

One failed required condition blocks the gate.

Common Release Mistakes

Mixing wet, dry, actual, standard and oxygen-corrected bases.
Averaging invalid analyzer data.
Treating detection-limit failures as harmless near a low permit limit.
Ignoring missing-data substitution rules.
Closing corrective action from a lower reading without maintenance records.
Comparing instantaneous data to rolling-average limits.
Treating CEMS valid-hour percentage as proof that high-emission event data are acceptable.
Using oxygen correction without confirming the oxygen analyzer validity for the same period.
Reporting control efficiency while ignoring the outlet permit limit.
Releasing operation before bypass cause, duration, notification and corrective action are closed.

Validation Package Checklist

Source mode, pollutant, averaging period and reporting basis.
Flow, concentration, moisture and oxygen correction records.
Analyzer calibration, drift, SNR and detection-limit evidence.
CEMS valid-hour, missing-data and rolling-average records.
Bypass, excursion and corrective-action records.
Permit condition, reviewer disposition and release authority.
Calibration gas certificate, expiration, traceability and span range.
Data acquisition flags showing valid, invalid, substituted and excluded periods.
Control-device operating status tied to the same averaging window.
Final disposition stating compliant, noncompliant, substituted, reported, corrected or held.

The final acceptance question is whether another reviewer could reproduce the same compliance conclusion from the records alone. If the calculation depends on undocumented basis choices or unflagged invalid data, the monitoring package should remain on hold.

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