8.1 Air Quality

Criteria Pollutants and WHO Guidelines

Regulatory agencies define criteria pollutants -- species for which ambient air quality standards are set based on health effects. The six criteria pollutants under the US Clean Air Act, along with WHO 2021 guideline values, are:

PM2.5 is the most lethal criteria pollutant globally. Particles with aerodynamic diameter less than 2.5 micrometers penetrate deep into the alveolar region of the lungs, enter the bloodstream, and cause systemic inflammation, atherosclerosis, and oxidative stress.

Gaussian Plume Dispersion Model

The Gaussian plume model is the foundational analytical solution for steady-state dispersion from a continuous point source. It assumes the concentration distribution is Gaussian in both the crosswind (y) and vertical (z) directions, with the plume advected downwind at mean wind speed u. For a source with effective stack height H (physical height + plume rise), the ground-level and elevated concentration is:

where:

• $Q$ = emission rate (g/s or kg/s)
• $u$ = mean wind speed at effective stack height (m/s)
• $\sigma_y(x)$, $\sigma_z(x)$ = horizontal and vertical dispersion parameters (m), functions of downwind distance x
• $H$ = effective stack height = physical height $h_s$ + plume rise $\Delta h$
• The second exponential term with $(z+H)$ represents the reflection of the plume from the ground (image source method)

At ground level (z = 0), the equation simplifies to:

The maximum ground-level concentration along the plume centerline (y = 0) occurs at the downwind distance where $\sigma_z = H/\sqrt{2}$. This is a critical regulatory metric for stack design.

Urban Box Model

A simple but useful approach for estimating urban-scale pollutant concentrations treats the city as a well-mixed box of volume V = L x W x H, where L is the city length along the wind, W is the crosswind width, and H is the mixing height. The mass balance yields:

where E is the total emission rate (g/s), $\tau = L/u$ is the ventilation time (flushing time), $k_c$ is a first-order chemical loss rate, and $C_{bg}$ is the background concentration. At steady state ($dC/dt = 0$):

The box model reveals that urban pollutant levels scale inversely with wind speed u and mixing height H -- explaining why stagnant conditions with low inversions produce the worst air quality episodes. The ventilation coefficient (u x H) is a key metric used operationally to forecast air quality.

Photochemical Smog and the NOₓ-VOC-O₃ Cycle

Photochemical smog is driven by the nonlinear interaction between nitrogen oxides (NOₓ = NO + NO₂) and volatile organic compounds (VOCs) in the presence of sunlight. The basic photostationary state cycle is:

This gives the Leighton relationship at photostationary state:

Without VOCs, this cycle simply recycles NO, NO₂, and O₃ with no net ozone production. However, VOCs react with OH radicals to produce peroxy radicals (RO₂, HO₂) that convert NO to NO₂ without consuming O₃:

This additional NO-to-NO₂ conversion pathway leads to net ozone accumulation. Urban ozone control is complicated because O₃ production can be either NOₓ-limited or VOC-limited depending on the NOₓ/VOC ratio. In VOC-limited regimes, reducing NOₓ alone can actually increase ozone -- a counterintuitive result with major policy implications.

Fortran: Urban Box Model for Pollutant Evolution

This Fortran program integrates the urban box model equation forward in time using an explicit Euler scheme, tracking pollutant concentration under time-varying emissions and a diurnally varying mixing height.

program box_model_pollution
  ! Urban box model for pollutant concentration evolution
  ! dC/dt = E/V - C/tau - k_c*C + C_bg/tau
  implicit none

  integer, parameter :: dp = selected_real_kind(15, 307)
  integer, parameter :: nsteps = 86400   ! 1-day simulation, dt=1s
  real(dp) :: C, E, L_city, W_city, H_mix, u_wind
  real(dp) :: tau, V, k_chem, C_bg, dt, time
  real(dp) :: hour, E_base, H_day, H_night
  integer  :: i, iunit

  ! --- Parameters ---
  L_city  = 20000.0_dp    ! City length along wind (m)
  W_city  = 15000.0_dp    ! City crosswind width (m)
  u_wind  = 3.0_dp        ! Mean wind speed (m/s)
  k_chem  = 1.0e-5_dp     ! Chemical loss rate (1/s)
  C_bg    = 5.0e-6_dp     ! Background concentration (g/m^3)
  E_base  = 500.0_dp      ! Base emission rate (g/s)
  H_day   = 1500.0_dp     ! Daytime mixing height (m)
  H_night = 200.0_dp      ! Nighttime mixing height (m)
  dt      = 1.0_dp        ! Time step (s)

  ! --- Initial condition ---
  C = C_bg   ! Start at background

  ! --- Output file ---
  iunit = 20
  open(unit=iunit, file='box_model_output.dat', status='replace')
  write(iunit, '(A)') '# Time(h)  Concentration(ug/m3)  MixHeight(m)  Emission(g/s)'

  ! --- Time integration ---
  do i = 1, nsteps
    time = real(i, dp) * dt
    hour = mod(time / 3600.0_dp, 24.0_dp)

    ! Diurnal mixing height: rises after sunrise, falls after sunset
    if (hour >= 7.0_dp .and. hour <= 18.0_dp) then
      H_mix = H_night + (H_day - H_night) * &
              sin(3.14159265_dp * (hour - 7.0_dp) / 11.0_dp)
    else
      H_mix = H_night
    end if

    ! Diurnal emission profile: peak during rush hours
    if (hour >= 7.0_dp .and. hour <= 9.0_dp) then
      E = E_base * 2.5_dp   ! Morning rush
    else if (hour >= 17.0_dp .and. hour <= 19.0_dp) then
      E = E_base * 2.0_dp   ! Evening rush
    else if (hour >= 9.0_dp .and. hour <= 17.0_dp) then
      E = E_base * 1.2_dp   ! Daytime
    else
      E = E_base * 0.3_dp   ! Nighttime
    end if

    ! Box model parameters
    V   = L_city * W_city * H_mix
    tau = L_city / u_wind

    ! Euler forward step
    C = C + dt * (E / V - C / tau - k_chem * C + C_bg / tau)
    if (C < 0.0_dp) C = 0.0_dp

    ! Write output every 5 minutes
    if (mod(i, 300) == 0) then
      write(iunit, '(F8.3, 2X, F12.4, 2X, F8.1, 2X, F8.1)') &
        hour, C * 1.0e6_dp, H_mix, E
    end if
  end do

  close(iunit)
  print *, 'Box model integration complete. Output: box_model_output.dat'

end program box_model_pollution