Air Quality and Control

A. Ambient and indoor air quality

Ambient Air Quality and Criteria Pollutants

Know the six criteria pollutants and the NAAQS, compute the Air Quality Index, and trace photochemical smog, ozone formation, acid deposition, and primary versus secondary pollutants.

Unlocks with an access pass — one-time payment, no auto-renew. View passes

Pass holders

C. Emissions

Emission Factors and Mass Balances

Estimate pollutant emission rates from AP-42 emission factors, close steady-state mass balances on processes and control devices, combine efficiencies in series, and convert ppm to mg/m³.

Unlocks with an access pass — one-time payment, no auto-renew. View passes

Pass holders

D. Atmospheric modeling and meteorology

The Gaussian Plume and Atmospheric Stability

Predict downwind ground-level concentrations from an elevated source using the Gaussian dispersion model, Pasquill stability classes, effective stack height, and the lapse rate.

Almost every air-dispersion question on the FE Environmental exam is a single equation wearing different costumes: the Gaussian plume. A continuous source — a stack, a flare, a vent — releases pollutant at rate

Q

, the wind carries it downwind at speed

u

, and turbulence spreads it into an ever-widening cone whose cross-section is a two-dimensional bell curve. Your job is almost always the same: find the concentration that a receptor on the ground actually breathes, some distance downwind. Points are lost not in the exponential but in three bookkeeping decisions — using the effective stack height instead of the physical one, reading the right dispersion coefficients

\sigma_y

and

\sigma_z

for the stability class, and keeping

Q

in

\mu\text{g/s}

so the answer lands in

\mu\text{g/m}^3

. This material lives in the FE Reference Handbook under Environmental Engineering, "Atmospheric Dispersion Modeling (Gaussian)."

The full Gaussian plume equation

The model treats the time-averaged plume as a Gaussian distribution in the crosswind (

y

) and vertical (

z

) directions superimposed on steady downwind advection. The centerline rides at the effective stack height

H

, and the ground at

z=0

acts as a mirror that reflects the plume back upward — that reflection is the second exponential term. Here

C

is the steady-state concentration at point

(x,y,z)

in

\mu\text{g/m}^3

,

Q

the emission rate in

\mu\text{g/s}

,

u

the wind speed at stack height in

\text{m/s}

, and

\sigma_y,\sigma_z

the horizontal and vertical dispersion parameters in meters, both functions of downwind distance

x

and stability class.

C = \dfrac{Q}{2\pi\, u\, \sigma_y \sigma_z}\, \exp\!\left(-\dfrac{y^2}{2\sigma_y^2}\right)\!\left[\exp\!\left(-\dfrac{(z-H)^2}{2\sigma_z^2}\right) + \exp\!\left(-\dfrac{(z+H)^2}{2\sigma_z^2}\right)\right]

The cases you actually compute: ground level and centerline

Two simplifications cover the vast majority of exam problems. First, the receptor is almost always at ground level,

z=0

, where the two reflection terms become identical and add to give a factor of two. Second, the worst-case receptor sits on the plume centerline,

y=0

, where the crosswind exponential is one. Imposing both at once collapses the full model into the compact ground-level, centerline concentration below — memorize this form, because it is the one you will substitute into most often.

C(x,0,0) = \dfrac{Q}{\pi\, u\, \sigma_y \sigma_z}\, \exp\!\left(-\dfrac{H^2}{2\sigma_z^2}\right)

Effective stack height and plume rise

The plume does not disperse from the top of the stack — buoyant, hot exhaust and exit momentum carry it higher before it levels off and begins to spread. The effective stack height

H

is the physical height

h

plus this plume rise

\Delta h

. Because

H

enters the ground-level equation inside a squared exponential, modest changes have an outsized effect: a taller effective stack pushes the centerline up and the ground-level concentration plummets. The handbook supplies

H = h + \Delta h

as a definition;

\Delta h

itself (from buoyancy and momentum flux, e.g. Holland's or Briggs' formulas) is generally given to you or stated in the problem.

H = h + \Delta h

Pasquill-Gifford stability classes and dispersion coefficients

Turbulence — and therefore

\sigma_y

and

\sigma_z

— is governed by atmospheric stability, sorted into six Pasquill classes from A (extremely unstable, strong daytime sun, light wind) through D (neutral, overcast or windy) to F (moderately stable, clear calm night). You select the class from a table keyed on surface wind speed, solar insolation by day, and cloud cover by night, then read

\sigma_y

and

\sigma_z

off the Pasquill-Gifford curves at your downwind distance

x

. The unstable classes spread a plume rapidly (large

\sigma

, low concentrations close in); the stable classes keep it tight and ribbon-like. The single most common error is mixing up classes or reading the curves at the wrong

x

. Remember: overcast forces Class D regardless of wind speed.

Maximum ground-level concentration

As you walk downwind, the ground-level centerline concentration first rises (the plume is bending down toward the ground) then falls (dilution wins). For neutral atmospheric conditions — the case the handbook prints — the peak occurs where the vertical spread reaches

\sigma_z = H/\sqrt{2}

; for other stability classes the handbook gives

(Cu/Q)_{\max}

and its distance on a chart instead. Substituting that condition into the ground-level equation makes the exponential equal

e^{-1}

, giving the clean maximum below. You evaluate

\sigma_y

at the same downwind distance where

\sigma_z = H/\sqrt{2}

. This tells a designer the highest concentration any ground receptor will ever see and roughly where it sits.

C_{\max} = \dfrac{Q}{\pi\, e\, u\, \sigma_y \sigma_z}\bigg|_{\,\sigma_z = H/\sqrt{2}}

Lapse rate, mixing height, and plume behavior

Stability ultimately comes from comparing the actual (environmental) lapse rate

\Gamma

— how fast ambient temperature falls with height — to the dry adiabatic lapse rate

\Gamma_{AD} = 0.98\,^\circ\text{C}

per

100\,\text{m}

(

5.4\,^\circ\text{F}

per

1{,}000\,\text{ft}

). If

\Gamma > \Gamma_{AD}

the air is unstable (a rising parcel stays warmer and keeps rising — looping plume); if

\Gamma = \Gamma_{AD}

it is neutral (coning); if

\Gamma < \Gamma_{AD}

it is stable (a rising parcel cools below its surroundings and sinks — fanning, poor dispersion). A temperature inversion (

\Gamma < 0

, temperature rising with height) is the extreme stable case and caps vertical mixing. The mixing height is the altitude of that cap; plumes trapped beneath it cannot vent upward, and a plume above an inversion that mixes down to the ground produces fumigation, the worst-case ground exposure.

\Gamma = -\dfrac{\Delta T}{\Delta z}; \qquad \Gamma > \Gamma_{AD}\ \text{unstable},\ \ \Gamma = \Gamma_{AD}\ \text{neutral},\ \ \Gamma < \Gamma_{AD}\ \text{stable}

Exam strategy

Work every plume problem in the same order. (1) Convert

Q

to

\mu\text{g/s}

immediately (

1\,\text{g/s}=10^6\,\mu\text{g/s}

) so the answer is in

\mu\text{g/m}^3

. (2) Find

H = h + \Delta h

— never plug in the physical stack height when a plume rise is given. (3) Fix the stability class from the wind/insolation table, then read

\sigma_y

and

\sigma_z

at the stated

x

. (4) Decide which simplification applies: ground-level centerline (

z=0,y=0

) is the default; only carry the crosswind term when

y\neq0

. For a maximum-concentration question in neutral conditions, use

\sigma_z=H/\sqrt{2}

and the

C_{\max}

form; for another stability class read

(Cu/Q)_{\max}

off the handbook's chart. Sanity check the exponent:

H^2/(2\sigma_z^2)

should be order one near the peak — a huge value means the plume has not yet reached the ground at that

x

.

Key equations

Gaussian plume (general)

C = \dfrac{Q}{2\pi u \sigma_y \sigma_z}\exp\!\left(-\dfrac{y^2}{2\sigma_y^2}\right)\!\left[\exp\!\left(-\dfrac{(z-H)^2}{2\sigma_z^2}\right)+\exp\!\left(-\dfrac{(z+H)^2}{2\sigma_z^2}\right)\right]

Concentration $C$ ( $\mu\text{g/m}^3$ ) at $(x,y,z)$ from a continuous elevated point source. $Q$ in $\mu\text{g/s}$ , $u$ in m/s, $\sigma_y,\sigma_z$ in m, $H$ effective stack height in m.

Ground-level, centerline concentration

C(x,0,0) = \dfrac{Q}{\pi u \sigma_y \sigma_z}\exp\!\left(-\dfrac{H^2}{2\sigma_z^2}\right)

Set $z=0$ (ground reflection doubles the term) and $y=0$ (on centerline). The everyday working form for worst-case ground exposure.

Effective stack height

H = h + \Delta h

$h$ = physical stack height (m), $\Delta h$ = plume rise (m) from buoyancy and exit momentum. Always use $H$ , not $h$ , in the dispersion equation.

Maximum ground-level concentration

C_{\max} = \dfrac{Q}{\pi e\, u\, \sigma_y \sigma_z}, \qquad \text{at } \sigma_z = \dfrac{H}{\sqrt{2}}

Peak ground concentration downwind for neutral atmospheric conditions (the handbook's stated case); the exponential becomes $e^{-1}$ when $\sigma_z=H/\sqrt2$ . Other stability classes: the handbook's $(Cu/Q)_{\max}$ chart. Evaluate $\sigma_y$ at that same $x$ .

Lapse rate

\Gamma = -\dfrac{\Delta T}{\Delta z}, \qquad \Gamma_{AD} = 0.98\,^\circ\text{C}/100\,\text{m}

Environmental lapse rate $\Gamma$ versus dry adiabatic $\Gamma_{AD}$ . $\Delta T$ = temperature change, $\Delta z$ = elevation change.

Stability criterion

\Gamma > \Gamma_{AD}\ \Rightarrow\ \text{unstable}; \quad \Gamma = \Gamma_{AD}\ \Rightarrow\ \text{neutral}; \quad \Gamma < \Gamma_{AD}\ \Rightarrow\ \text{stable}

Inversion ( $\Gamma<0$ ) is the strongly stable limit that caps mixing and traps the plume.

ppb / concentration conversion

\text{ppb} = \dfrac{(\mu\text{g/m}^3)\,RT}{P\,\text{MW}}

Convert mass concentration to volume mixing ratio. $R=0.0821\,\text{L·atm/(mol·K)}$ , $T$ in K, $P$ in atm, MW in g/mol.

Worked examples

Ground-level concentration downwind of a stack

Problem. A power-plant stack emits $\text{SO}_2$ at $Q=80\,\text{g/s}$ . Wind speed at stack height is $u=5\,\text{m/s}$ and the effective stack height is $H=50\,\text{m}$ . Under neutral (Class D) conditions, at $x=1\,\text{km}$ downwind the Pasquill-Gifford curves give $\sigma_y=68\,\text{m}$ and $\sigma_z=33\,\text{m}$ . Find the ground-level concentration on the plume centerline.

Solution. Convert the source:

Q = 80\,\text{g/s} = 8.0\times10^{7}\,\mu\text{g/s}

. Use the ground-level centerline form (

z=0,\,y=0

):

C = \frac{Q}{\pi u \sigma_y \sigma_z}\exp\!\left(-\frac{H^2}{2\sigma_z^2}\right).

Exponent:

\dfrac{H^2}{2\sigma_z^2} = \dfrac{50^2}{2(33)^2} = \dfrac{2500}{2178} = 1.148

, so

e^{-1.148}=0.317

. Prefactor:

\pi u \sigma_y \sigma_z = \pi (5)(68)(33) = 35{,}250\,\text{m}^3/\text{s}

.

C = \frac{8.0\times10^{7}}{35{,}250}\,(0.317) = (2270)(0.317) = 720\ \mu\text{g/m}^3.

**Answer:

C \approx 720\ \mu\text{g/m}^3

.** Units check:

(\mu\text{g/s})/(\text{m}^3/\text{s}) = \mu\text{g/m}^3

. The exponent is order one, confirming the plume has reached the ground at this distance.

C = \dfrac{8.0\times10^{7}}{\pi(5)(68)(33)}\,e^{-1.148} \approx 720\ \mu\text{g/m}^3

Maximum ground-level concentration

Problem. For the same stack ( $Q=80\,\text{g/s}$ , $u=5\,\text{m/s}$ , $H=50\,\text{m}$ , Class D), find the maximum ground-level concentration. At the downwind distance where $\sigma_z=H/\sqrt2$ (just beyond $1\,\text{km}$ for Class D), the curves give $\sigma_y\approx70\,\text{m}$ .

Solution. The maximum occurs where the vertical spread satisfies

\sigma_z = H/\sqrt2 = 50/1.414 = 35.4\,\text{m}

, which makes the exponential exactly

e^{-1}

.

C_{\max} = \frac{Q}{\pi e\, u\, \sigma_y \sigma_z}.

With

Q=8.0\times10^{7}\,\mu\text{g/s}

:

C_{\max} = \frac{8.0\times10^{7}}{\pi (2.718)(5)(70)(35.4)} = \frac{8.0\times10^{7}}{1.057\times10^{5}} = 757\ \mu\text{g/m}^3.

**Answer:

C_{\max} \approx 757\ \mu\text{g/m}^3

.** Sanity check: this correctly exceeds the

720\ \mu\text{g/m}^3

found at

x=1\,\text{km}

in the previous example — as it must, since the peak sits slightly farther downwind, where

\sigma_z

has grown from

33

to

35.4\,\text{m}

while

\sigma_y

is still near

70\,\text{m}

. A computed maximum that fell below an already-known receptor value would flag an inconsistent

\sigma_y

. The procedure — set

\sigma_z=H/\sqrt2

, then read

\sigma_y

at that same distance — is the exam method.

C_{\max} = \dfrac{8.0\times10^{7}}{\pi e (5)(70)(35.4)} \approx 757\ \mu\text{g/m}^3

Stability class from the lapse rate

Problem. On a given morning the air temperature is $22.0\,^\circ\text{C}$ at the $2\,\text{m}$ reference height and $20.0\,^\circ\text{C}$ at $302\,\text{m}$ . Determine the environmental lapse rate, classify the stability, and describe the expected plume behavior.

Solution. Take both differences in the same direction, top minus bottom:

\Delta T = T_{\text{top}} - T_{\text{bottom}} = 20.0 - 22.0 = -2.0\,^\circ\text{C}

over

\Delta z = 302 - 2 = 300\,\text{m}

(temperature decreases with height).

\Gamma = -\frac{\Delta T}{\Delta z} = -\frac{-2.0\,^\circ\text{C}}{300\,\text{m}} = 0.667\,^\circ\text{C}/100\,\text{m}.

Compare to the dry adiabatic rate

\Gamma_{AD}=0.98\,^\circ\text{C}/100\,\text{m}

. Since

\Gamma = 0.667 < 0.98 = \Gamma_{AD}

, the atmosphere is **stable**. **Answer:

\Gamma = 0.667\,^\circ\text{C}/100\,\text{m}

, stable.** A stable layer suppresses vertical mixing, so the plume disperses laterally but barely vertically — a flat, ribbon-like ·fanning· plume with poor ground-level dispersion near the source. Sanity check: temperature still decreases with height (no inversion), so this is stable but not the trapped inversion limit.

\Gamma = -\dfrac{\Delta T}{\Delta z} = -\dfrac{-2.0\,^\circ\text{C}}{300\,\text{m}} = 0.667\,^\circ\text{C}/100\,\text{m} < \Gamma_{AD}

Common pitfalls

•Using the physical stack height $h$ instead of the effective height $H=h+\Delta h$ . Plume rise can easily double the effective height and cut ground concentration by an order of magnitude — always add $\Delta h$ first.
•Leaving $Q$ in g/s. The handbook Gaussian equation expects $Q$ in $\mu\text{g/s}$ to give $C$ in $\mu\text{g/m}^3$ ; multiply g/s by $10^6$ before substituting.
•Forgetting the factor of two at the ground. The general equation has two reflection terms; at $z=0$ they are equal and add, which is why the ground-level form has $\pi$ (not $2\pi$ ) in the denominator.
•Reading $\sigma_y,\sigma_z$ at the wrong distance or wrong stability class. They depend on both $x$ and class; an overcast sky forces Class D regardless of wind speed.
•Confusing mass concentration ( $\mu\text{g/m}^3$ ) with volume mixing ratio (ppm/ppb). Convert with $\text{ppb}=(\mu\text{g/m}^3)RT/(P\cdot\text{MW})$ — never equate them directly.
•Sign error in the lapse rate: a stable atmosphere has $\Gamma<\Gamma_{AD}$ (small or negative slope), an unstable one $\Gamma>\Gamma_{AD}$ . An inversion ( $T$ rising with height) is strongly stable, not unstable.
•Assuming the maximum concentration is at the receptor you were given. The peak is at $\sigma_z=H/\sqrt2$ ; you must read $\sigma_y$ at that location, not at an arbitrary downwind distance.

References

NCEES FE Reference Handbook — Environmental Engineering
Turner, Workbook of Atmospheric Dispersion Estimates — Pasquill-Gifford curves and stability classification
Davis & Cornwell, Introduction to Environmental Engineering — air dispersion chapter

F. Particle treatment technologies

Particulate Control: Cyclones, ESPs, and Baghouses

Size and rate the three workhorse particulate collectors — cyclone cut diameter, the ESP Deutsch-Anderson equation, and baghouse air-to-cloth ratio — and match each device to particle size.

Unlocks with an access pass — one-time payment, no auto-renew. View passes

Pass holders

G. Indoor air quality modeling and controls

Indoor Air Quality Modeling and Ventilation

Model indoor pollutant concentrations with the well-mixed mass balance, find the steady state, track transient build-up and decay through the air-exchange rate, and size ventilation.

Unlocks with an access pass — one-time payment, no auto-renew. View passes

Pass holders