7.3 Atmospheric Dispersion Modeling & Indoor Air Quality

Atmospheric Dispersion Modeling & Indoor Air Quality (IAQ)

Once gaseous or particulate air pollutants are discharged from elevated industrial stacks or fugitive indoor sources, their movement is governed by atmospheric transport (advection) and turbulent mixing (dispersion). Environmental engineers utilize steady-state Gaussian plume dispersion models to evaluate outdoor downwind ambient concentrations against National Ambient Air Quality Standards (NAAQS), while applying indoor mass balance equations to ensure acceptable Indoor Air Quality (IAQ) under ASHRAE 62.1 standards.

Gaussian Plume Dispersion Model Derivation

The EPA regulatory baseline model assumes continuous emission from a point source into an atmosphere with constant wind speed ($u$) along the $x$-axis. Dispersion in the crosswind ($y$) and vertical ($z$) directions follows double-Gaussian normal probability distributions.

Full Steady-State Gaussian Equation

Including total reflection of the plume at the impenetrable ground surface ($z=0$), the downwind ground-level or elevated concentration $C(x,y,z;H)$ ($g/m^3$ or $\mu g/m^3$) is calculated as:

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

Where:

• $Q$ = Mass emission rate of pollutant ($g/s$)
• $u$ = Mean wind speed at effective stack height ($m/s$)
• $\sigma_y, \sigma_z$ = Standard deviations of plume concentration distribution in crosswind and vertical directions ($m$). These grow as a function of downwind distance ($x$) and Pasquill-Gifford stability class.
• $H$ = Effective stack release height ($m$)
• $y$ = Crosswind distance from plume centerline ($m$)
• $z$ = Vertical receptor height above ground ($m$)

Ground-Level Centerline Concentration

For a receptor located directly on the ground ($z=0$) along the plume centerline ($y=0$), the equation simplifies to:

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

Maximum Ground-Level Concentration Location

The maximum ground-level concentration occurs downwind where $\sigma_z(x) = \frac{H}{\sqrt{2}} \approx 0.707 H$:

$C_{max} = \frac{2 Q}{\pi e u H^2 \left(\frac{\sigma_y}{\sigma_z}\right)}$

Plume Rise Thermodynamics & Effective Stack Height

The effective stack height ($H$) accounts for both the physical stack height ($h_s$) and the thermal/momentum plume rise ($\Delta h$):

$H = h_s + \Delta h$

Holland's Plume Rise Formula

For neutral stability conditions, plume rise is estimated via Holland's equation:

$\Delta h = \frac{v_s d}{u} \left[ 1.5 + 2.68 \times 10^{-3} P \cdot d \left(\frac{T_s - T_a}{T_s}\right) \right]$

Where $v_s$ is stack exit gas velocity ($m/s$), $d$ is inner stack diameter ($m$), $P$ is atmospheric pressure ($mb$), $T_s$ is stack gas temperature ($K$), and $T_a$ is ambient air temperature ($K$).

Briggs Buoyancy Flux Parameter

For hot, buoyant plumes, Briggs equations use the Buoyancy Flux ($F$):

$F = g v_s d^2 \left(\frac{T_s - T_a}{4 T_s}\right)$

Pasquill-Gifford Stability Classes & Plume Behavior

Atmospheric stability is classified from Class A (Extremely Unstable) to Class F (Moderately Stable) based on surface wind speed and solar radiation.

Environmental Lapse Rate (ELR) vs. Dry Adiabatic Lapse Rate (DALR)

Stability depends on comparing the actual ELR ($-\frac{dT}{dz}$) with the DALR ($\Gamma_{d} = 9.8^\circ C/km$ or $5.4^\circ F/1000 \ ft$):

• Unstable (Superadiabatic, $ELR > 9.8^\circ C/km$): Strong vertical mixing. Produces Looping plumes (Class A/B).
• Neutral (Neutral, $ELR = 9.8^\circ C/km$): Moderate mixing. Produces Coning plumes (Class D, overcast day/night).
• Stable (Inversion / Subadiabatic, $ELR < 9.8^\circ C/km$): Suppressed vertical mixing. Produces Fanning plumes (Class E/F, clear night).
• Lofting Plume: Inversion below stack, unstable above. Highly desirable; traps emissions above ground.
• Fumigation Plume: Unstable below stack, inversion aloft (e.g., morning solar heating breaking ground inversion). Dangerous; brings high concentration plume to ground level across a wide area.

Indoor Air Quality (IAQ) Dynamic Mass Balances

Indoor air pollution inside commercial and residential buildings is evaluated using single-zone or multi-zone mass balance models.

Dynamic Governing Equation

$V \frac{dC}{dt} = G + Q C_{in} - Q C - k V C - E Q_{rec} C$

Where:

• $V$ = Enclosed room volume ($m^3$)
• $C(t)$ = Indoor pollutant concentration ($mg/m^3$ or $ppm$)
• $G$ = Indoor generation rate ($mg/h$)
• $Q$ = Outdoor fresh air ventilation rate ($m^3/h$)
• $C_{in}$ = Outdoor makeup air pollutant concentration ($mg/m^3$)
• $k$ = First-order decay / surface deposition rate constant ($h^{-1}$)
• $E$ = Air cleaner filtration efficiency ($0 - 1.0$)
• $Q_{rec}$ = Recirculated filtration air flow rate ($m^3/h$)

Steady-State Solution ($dC/dt = 0$)

$C_{ss} = \frac{G + Q C_{in}}{Q + k V + E Q_{rec}}$

If there is no internal decay ($k=0$) and no recirculated filtration ($Q_{rec}=0$):

$C_{ss} = C_{in} + \frac{G}{Q}$

ASHRAE Standard 62.1 Ventilation Requirements

ASHRAE Standard 62.1 specifies minimum outdoor airflow rates ($Q_{tot}$) based on human occupant density ($P_z$) and floor area ($A_z$):

$Q_{tot} = R_p P_z + R_a A_z$

Where $R_p$ is outdoor air rate per person ($L/s\cdot person$ or $cfm/person$) and $R_a$ is outdoor air rate per area ($L/s\cdot m^2$ or $cfm/ft^2$). Human bio-effluent $CO_2$ generation ($G_{CO2} \approx 0.005 \ L/s \cdot person$) is used as a surrogate indicator for adequate ventilation, maintaining indoor $CO_2 < 1000 \ ppmv$ ($< 700 \ ppmv$ above ambient).