Formula sheet

Air Quality and Emissions Control Systems Formula Sheet

Air-emissions formulas for stack mass rate, gas correction, capture flow, pressure drop, fan power, control efficiency, media capacity, bypass load, uncertainty, and compliance.

Branch: Environmental Engineering
Content: Formula sheet
Updated: Jun 23, 2026
Revision: v1.0.0 · reviewed

This formula sheet collects first-pass calculations used in air quality and emissions control engineering. Use it to screen stack mass rates, local exhaust capture, duct velocity, pressure drop, fan power, particulate and vapor control efficiency, media capacity, bypass load, monitoring uncertainty, and release margins.

These equations support engineering judgement. They do not replace approved stack-test methods, permit-specific reference conditions, dispersion modelling, ventilation standards, process safety review, control-device vendor limits, or site operating procedures.

Basis and Conventions

State the basis before calculating:

actual, standard, normal, wet, dry, or oxygen-corrected gas volume;
averaging period, operating mode, production rate, fuel, material feed, and control-device state;
concentration units, such as $\text{mg}/\text{m}^3$ , $\text{ppm}_v$ , $\text{g}/\text{dscm}$ , or $\text{kg}/\text{h}$ ;
whether the boundary is source generation, hood capture, duct transport, control-device inlet, stack outlet, bypass, or receptor exposure;
whether the result is for compliance reporting, design sizing, maintenance action, alarm response, troubleshooting, or validation.

The most common error is mixing gas bases. A concentration on a dry standard basis should not be multiplied by an actual wet flow unless the flow has been converted to the same basis.

Symbols

Symbol	Meaning	Typical unit
$Q$	volumetric gas flow rate on the stated basis	$\text{m}^3/\text{s}$
$v$	average gas velocity	$\text{m}/\text{s}$
$A$	flow, hood, or filter area	$\text{m}^2$
$\rho$	gas density	$\text{kg}/\text{m}^3$
$C$	pollutant concentration on the stated basis	$\text{mg}/\text{m}^3$
$\dot{M}$	pollutant mass rate	$\text{kg}/\text{h}$ or $\text{g}/\text{s}$
$M$	total pollutant mass	$\text{kg}$
$\Delta p$	pressure drop or fan static pressure requirement	$\text{Pa}$
$P$	power	$\text{W}$
$\eta$	efficiency or removal fraction	dimensionless
$D$	duct or stack diameter	$\text{m}$
$\mu$	dynamic viscosity	$\text{Pa}\cdot\text{s}$
$u_c$	combined standard uncertainty	same as measured quantity
$k$	coverage factor for a decision margin	dimensionless

Gas Flow and Reference Basis

Volumetric flow from velocity:

Q=vA

Circular duct or stack area:

\displaystyle A=\frac{\pi D^2}{4}

Mass flow of the gas stream:

\dot{m}=\rho Q

Ideal-gas conversion from actual to standard volume flow:

\displaystyle Q_{std}=Q_{act}\frac{p_{act}}{p_{std}}\frac{T_{std}}{T_{act}}

Use absolute pressure and absolute temperature. This conversion assumes the same molar flow, ideal-gas behaviour, and no condensation or leakage between the measurement basis and the reporting basis.

Dry-basis correction from wet-basis concentration:

\displaystyle C_{dry}=\frac{C_{wet}}{1-B_w}

where $B_w$ is the water-vapour volume fraction. This correction is invalid if moisture was not measured representatively or if condensation changed the pollutant concentration before sampling.

Oxygen correction for combustion exhaust when required by the reporting method:

\displaystyle C_{ref}=C_{meas}\frac{20.9-O_{2,ref}}{20.9-O_{2,meas}}

where oxygen values are in percent by volume on a dry basis. Apply this only when the permit or test method specifies an oxygen reference. It should not be used to hide dilution air, sampling leakage, or a changed combustion condition.

Pollutant Mass Rate and Total Load

Pollutant mass rate:

\dot{M}=QC

If $Q$ is in $\text{m}^3/\text{s}$ and $C$ is in $\text{mg}/\text{m}^3$ :

\dot{M}_{kg/h}=3.6\times10^{-3}QC

Total load over a time interval:

M=\int_0^T Q(t)C(t)\,dt

Discrete approximation:

M\approx \sum_i Q_iC_i\Delta t_i

For the discrete form, use $C$ in $\text{kg}/\text{m}^3$ if $Q$ is in $\text{m}^3/\text{s}$ and $\Delta t$ is in seconds. If concentration varies with production cycle, startup, cleaning pulse, or batch peak, a simple arithmetic mean can understate the total load.

Emission factor estimate:

\dot{M}=EF\dot{R}

where $EF$ is an emission factor and $\dot{R}$ is production, fuel, material, or activity rate. Emission factors are screening tools; measured source data are stronger when the process condition is representative.

Capture and Local Exhaust Flow

Required hood or face flow:

Q_{req}=v_fA_h

Capture-flow margin:

M_Q=Q_{meas}-Q_{req}

Relative capture shortfall:

\displaystyle S_Q=\frac{Q_{req}-Q_{meas}}{Q_{req}}

Use $S_Q$ only when $Q_{meas}<Q_{req}$ . Face velocity is a rough screen. Actual capture depends on hood geometry, source distance, cross-drafts, thermal plumes, make-up air, enclosure leakage, worker position, and duct balance.

Duct or stack velocity:

\displaystyle v=\frac{Q}{A}

Duct Reynolds number:

\displaystyle Re=\frac{\rho vD}{\mu}

For particulate transport, low velocity can allow settling and blockage. Very high velocity can cause erosion, noise, vibration, and unnecessary fan energy. The acceptable velocity range depends on particle size, density, moisture, explosibility, duct layout, and maintenance access.

Pressure Drop and Fan Power

Fan input power:

\displaystyle P_{in}=\frac{Q\Delta p}{\eta_{fm}}

where $\eta_{fm}$ is combined fan, drive, and motor efficiency.

Pressure-drop margin to an alarm or maintenance trigger:

M_{\Delta p}=\Delta p_{trigger}-\Delta p_{meas}

Pressure-drop trend:

\displaystyle r_{\Delta p}=\frac{\Delta p_2-\Delta p_1}{t_2-t_1}

A rising pressure drop can indicate filter loading, wet cake, plugged demisters, fouled scrubber packing, damper movement, duct deposits, or an operating point shift. A falling pressure drop can be just as important if it indicates broken bags, bypass, fan slip, duct leakage, or low process flow.

Control-Device Efficiency

Removal efficiency from inlet and outlet concentration:

\displaystyle \eta_c=\frac{C_{in}-C_{out}}{C_{in}}

Outlet concentration from inlet concentration:

C_{out}=C_{in}(1-\eta_c)

Outlet mass rate:

\dot{M}_{out}=\dot{M}_{in}(1-\eta_c)

Required efficiency to meet an outlet concentration limit:

\displaystyle \eta_{req}=1-\frac{C_{limit}}{C_{in}}

Overall efficiency for independent stages in series:

\eta_{total}=1-\prod_i(1-\eta_i)

Do not use a single overall efficiency when particle-size efficiency, chemical speciation, humidity, temperature, or concentration peaks control the decision. For hazardous compounds, destruction/removal efficiency may need a pollutant-specific method.

Fabric Filters and Particulate Collection

Air-to-cloth ratio:

\displaystyle ACR=\frac{Q}{A_{cloth}}

Captured dust rate:

\dot{M}_{dust}=Q(C_{in}-C_{out})

If $Q$ is in $\text{m}^3/\text{s}$ and concentration is in $\text{mg}/\text{m}^3$ :

\dot{M}_{dust,kg/h}=3.6\times10^{-3}Q(C_{in}-C_{out})

Screening time to a dust-capacity or handling limit:

\displaystyle t_{cap}=\frac{M_{cap}}{\dot{M}_{dust}}

The capacity may be a hopper cleanout limit, process hold point, waste container limit, or conservative media-loading limit. It is not automatically the same as filter life. Moisture, oil mist, sticky dust, pulsing pressure, bag damage, hopper plugging, and combustible-dust safeguards can dominate the actual maintenance interval.

Adsorption, Scrubbing, and Thermal Oxidation

Adsorbent service-life screen:

\displaystyle t_{bed}=\frac{f_u q_u M_{media}}{\dot{M}_{in}}

where $q_u$ is usable pollutant capacity per media mass and $f_u$ is a conservative utilization factor. Breakthrough can occur earlier because of humidity, temperature, competing compounds, channeling, concentration peaks, media aging, and poor flow distribution.

Liquid-to-gas ratio for a wet scrubber:

\displaystyle \frac{L}{G}=\frac{Q_L}{Q_G}

Use consistent volume or mass flow bases. Scrubber performance also depends on gas-liquid contact, droplet size, packing condition, reagent strength, solubility, reaction rate, pressure drop, mist elimination, scaling, and wastewater handling.

Reagent stoichiometric ratio:

\displaystyle SR=\frac{n_{reagent,supplied}}{n_{reagent,stoich}}

Residence time in a thermal or catalytic oxidizer:

\displaystyle \tau=\frac{V_{hot}}{Q_{hot}}

Destruction and removal efficiency:

\displaystyle DRE=\frac{\dot{M}_{in}-\dot{M}_{out}}{\dot{M}_{in}}

Sensible heating duty:

\dot{Q}_{heat}=\dot{m}c_p(T_{out}-T_{in})

Fuel or utility input with heat recovery:

\displaystyle \dot{Q}_{fuel}\approx\frac{\dot{m}c_p(T_{set}-T_{in})(1-\varepsilon_{HR})}{\eta_{burner}}

This is a screening relation. Real oxidizer sizing needs lower flammability limit review, oxygen, mixing, residence-time distribution, catalyst condition, temperature uniformity, safety interlocks, startup logic, and emergency venting.

Bypass and Abnormal Operation

Average mass rate with bypass fraction:

\dot{M}_{avg}=f_b\dot{M}_{uncontrolled}+(1-f_b)\dot{M}_{controlled}

Event mass during a bypass or upset:

M_{event}=\dot{M}_{event}t_{event}

Incremental mass above normal controlled operation:

M_{extra}=(\dot{M}_{event}-\dot{M}_{controlled})t_{event}

Bypass calculations are decision support, not permission. A defensible bypass record should state the reason, duration, source condition, control path affected, monitoring state, approval, compensating action, and evidence that normal control was restored.

Monitoring and Uncertainty

Signal-to-noise ratio:

\displaystyle SNR=\frac{S}{N}

Combined standard uncertainty from independent components:

u_c=\sqrt{\sum_i u_i^2}

Upper decision value for a measured concentration:

C_{upper}=C_{meas}+ku_c

Compliance or release guard margin:

G=C_{limit}-C_{upper}

If $G>0$ , the measured value is below the limit with the selected uncertainty allowance. If $G\le 0$ , the result is too close to the limit or above it; investigate measurement quality, operating condition, and corrective action before treating the condition as released.

Data capture fraction:

\displaystyle F_{data}=\frac{t_{valid}}{t_{required}}

Monitoring evidence should include calibration, sampling line condition, flow profile, moisture control, analyzer range, response time, interference, data validation rules, and the process state represented by the measurement.

Reliability and Interlock Screening

Availability from mean time between failures and mean time to repair:

\displaystyle A=\frac{MTBF}{MTBF+MTTR}

Expected uncontrolled time over an operating period:

t_{uncontrolled}=(1-A)t_{operating}

Risk priority number:

RPN=S\cdot O\cdot D

where $S$ is severity, $O$ is occurrence, and $D$ is detection score under the selected site ranking. Use RPN only as a prioritization aid; high-severity failure modes may require action even with a moderate numerical score.

Production interlock margin:

M_I=x_{meas}-x_{trip}

where $x$ may be capture flow, fan status, differential pressure band, reagent flow, oxidizer temperature, carbon-bed breakthrough indicator, damper position, or analyzer validity. The sign convention must be stated. An interlock should trip on the variable that proves control readiness, not merely on equipment power.

Worked Screening Example

A solids-handling line exhausts to a baghouse. The operating data are:

Quantity	Value
stack flow on the reporting basis	$Q=4.2\ \text{m}^3/\text{s}$
baghouse inlet particulate concentration	$C_{in}=750\ \text{mg}/\text{m}^3$
baghouse outlet particulate concentration	$C_{out}=18\ \text{mg}/\text{m}^3$
outlet concentration limit	$C_{limit}=25\ \text{mg}/\text{m}^3$
oxygen at measurement condition	$O_{2,meas}=9\%$
reference oxygen	$O_{2,ref}=7\%$
total pressure drop	$\Delta p=1200\ \text{Pa}$
fan and motor efficiency	$\eta_{fm}=0.63$
filter cloth area	$A_{cloth}=240\ \text{m}^2$
hopper cleanout screening capacity	$M_{cap}=350\ \text{kg}$
combined standard uncertainty of outlet concentration	$u_c=2.5\ \text{mg}/\text{m}^3$
coverage factor for release check	$k=2$
possible bypass duration	$10\ \text{min/day}$

Step 1: Removal Efficiency

\displaystyle \eta_c=\frac{750-18}{750}=0.976=97.6\%

Required efficiency at the same inlet concentration is:

\displaystyle \eta_{req}=1-\frac{25}{750}=0.9667=96.7\%

The measured removal efficiency is above the simplified requirement, but this does not yet prove compliance because gas basis, monitoring uncertainty, and bypass state still matter.

Step 2: Stack Mass Rates

Controlled outlet mass rate:

\dot{M}_{out}=3.6\times10^{-3}(4.2)(18)=0.272\ \text{kg/h}

Uncontrolled inlet mass rate:

\dot{M}_{in}=3.6\times10^{-3}(4.2)(750)=11.34\ \text{kg/h}

The baghouse is reducing the particulate mass rate by about:

11.34-0.272=11.07\ \text{kg/h}

This difference is also the approximate captured dust rate under the stated basis.

Step 3: Oxygen-Corrected Outlet Concentration

If the permit limit is stated at $7\%$ oxygen on a dry basis:

\displaystyle C_{ref}=18\frac{20.9-7}{20.9-9}

\displaystyle C_{ref}=18\frac{13.9}{11.9}=21.0\ \text{mg}/\text{m}^3

The oxygen-corrected value remains below the $25\ \text{mg}/\text{m}^3$ concentration limit. This calculation is valid only if both the concentration and oxygen values are on the required dry basis and measured under the applicable method.

Step 4: Fan Power

\displaystyle P_{in}=\frac{Q\Delta p}{\eta_{fm}}

\displaystyle P_{in}=\frac{4.2(1200)}{0.63}=8000\ \text{W}=8.0\ \text{kW}

The value is a useful operating screen, but the fan curve should still be checked. Filter loading or damper movement can shift the operating point while the fan continues to run.

Step 5: Air-to-Cloth Ratio and Dust Capacity

\displaystyle ACR=\frac{4.2}{240}=0.0175\ \text{m}/\text{s}

Captured dust rate:

\dot{M}_{dust}=3.6\times10^{-3}(4.2)(750-18)=11.07\ \text{kg/h}

Screening time to the hopper cleanout limit:

\displaystyle t_{cap}=\frac{350}{11.07}=31.6\ \text{h}

This does not mean the bags can be ignored for 31.6 hours. It means the selected dust-handling capacity reaches its screening limit after about 32 operating hours at this load unless cleaning, discharge, or production scheduling keeps the system within its envelope.

Step 6: Monitoring Guard Margin

Upper decision value:

C_{upper}=C_{ref}+ku_c

C_{upper}=21.0+2(2.5)=26.0\ \text{mg}/\text{m}^3

Guard margin:

G=25.0-26.0=-1.0\ \text{mg}/\text{m}^3

The nominal corrected concentration is below the limit, but the uncertainty-adjusted decision margin is negative. A conservative release decision would require better measurement evidence, repeated data, lower process load, improved control condition, or a formal rule that permits a different decision basis.

Step 7: Bypass Load

Convert bypass time:

\displaystyle t_b=\frac{10}{60}=0.1667\ \text{h}

Uncontrolled bypass mass:

M_b=11.34(0.1667)=1.89\ \text{kg/day}

Mass that would have been emitted if normal control remained active during the same time:

M_c=0.272(0.1667)=0.045\ \text{kg/day}

Incremental bypass mass:

M_{extra}=1.89-0.045=1.85\ \text{kg/day}

A ten-minute bypass can dominate the daily particulate load. The engineering decision should therefore treat bypass state as a compliance-critical condition, not as a maintenance detail.

Engineering Interpretation

The device appears effective by nominal removal efficiency and concentration. The release decision is still not automatic because the uncertainty-adjusted concentration margin is negative and the possible bypass event is large compared with the controlled stack rate.

The defensible next steps are to confirm the reporting basis, repeat or improve monitoring evidence, verify fan and differential-pressure stability, inspect bypass dampers, ensure hopper cleanout capacity, and document that production cannot run uncontrolled without a managed approval path and emissions record.

Common Mistakes

Common mistakes include multiplying a dry concentration by a wet actual flow, using average concentration without flow weighting, treating fan running status as proof of capture, reporting removal efficiency without inlet basis, and ignoring bypass time because it is short.

Other frequent mistakes are applying oxygen correction to a non-combustion source, treating a pressure-drop alarm as only a maintenance issue, sizing carbon beds from ideal capacity with no humidity or competing-compound allowance, using a single overall efficiency for a particle-size problem, and declaring compliance when the measurement uncertainty overlaps the limit.

Practical Validation Checklist

Before relying on an air-emissions calculation, confirm that:

the gas flow and concentration are on the same basis;
the operating condition matches the design, test, or permit basis;
capture flow, duct velocity, pressure drop, and fan point are plausible together;
control-device efficiency is supported by inlet and outlet evidence;
media capacity, reagent supply, hopper handling, and waste streams are within limits;
bypass, damper, interlock, and maintenance states are recorded;
monitoring uncertainty and data capture support the decision being made;
corrective-action closeout includes post-repair evidence under the condition that created the risk.

A strong calculation does more than produce a number. It preserves the chain from source condition to control boundary, measured evidence, uncertainty, operating envelope, and release decision.

REF

Disciplines