# 19.8: Review

- Page ID
- 10860

Pollutants emitted from a smoke stack will blow downwind and disperse by turbulent mixing with ambient air. By designing a stack of sufficient height, pollutants at ground level can become sufficiently dilute as to not exceed local environmental air-quality standards. Additional buoyant plume rise above the physical stack top can further reduce ground-level concentrations.

Air-quality standards do not consider instantaneous samples of pollutant concentration. Instead, they are based on measured averages over time. To model such averages, statistical descriptions of dispersion must be used, including the center of mass (plume centerline) and the standard deviation of location (proportional to plume spread).

For emissions in the boundary layer, the amount of dispersion depends on the type of turbulence. This relationship can be described by Taylor’s statistical theory.

During daytime conditions of free convection, thermals cause a peculiar form of dispersion that often brings high concentrations of pollutants close to the ground. At night, turbulence is suppressed in the vertical, causing little dispersion. As pollutants remain aloft for this case, there is often little hazard at ground level. Turbulent dispersion is quite anisotropic for these convective and stable cases.

Science Graffito

“The service we render to others is really the rent we pay for our room on the Earth.”

– Sir Wilfred Grenfell.

**Sample Application**

Source emissions of 200 g s^{–1} of SO_{2} occur at height 150 m. The environment is statically unstable, with a Deardorff convective velocity of 1 m s^{–1}, a mixed layer depth of 600 m, and a mean wind speed of 4 m s^{–1}.

Find the concentration at the ground 3 km downwind from the source, directly beneath the plume centerline.

**Find the Answer**

Given: Q = 200 g s^{–1}, z_{s} = 150 m, z_{i} = 600 m, M = 4 m s^{–1}, w_{*} = 1 m s^{–1}

Find: c (µg m^{–3}) at x = 3 km, y = z = 0.

Use eq. (19.25):

\(\ X=\frac{(3000 \mathrm{m}) \cdot(1 \mathrm{m} / \mathrm{s})}{(600 \mathrm{m}) \cdot(4 \mathrm{m} / \mathrm{s})}=1.25\)

Use eq. (19.29): Z_{s} = (150m) / (600m) = 0.25

From Fig. 19.8c, read C_{y} ≈ 0.9 at X = 1.25 and Z = 0.

Use eq. (19.38): σ_{yd} ≈ 0.5 · (1.25) = 0.625

Use eq. (19.37) with Y = 0:

\(\ C=\frac{0.9}{\sqrt{2 \pi} \cdot 0.625}=0.574\)

Finally, use eq. (19.23) rearranging it first to solve for concentration in physical units:

\(\ c=\frac{C \cdot Q}{z_{i}^{2} \cdot M}=\frac{(0.574) \cdot(200 \mathrm{g} / \mathrm{s})}{(600 \mathrm{m})^{2} \cdot(4 \mathrm{m} / \mathrm{s})}=79.8 \mu \mathrm{g} \mathrm{m}^{-3}\)

**Check:** Units OK. Physics OK.

**Exposition:** We were lucky that the dimensionless source height was 0.25, which allowed us to use Fig. 19.8c. For other source heights not included in that figure, we would have to create new figures using equations (19.33) through (19.36).

In statically neutral conditions of overcast skies and strong winds, turbulence is more isotropic. Smoke plumes disperse at roughly equal rates in the vertical and lateral directions, and are well described by Gaussian formulae.

Various classification schemes have been designed to help determine the appropriate characteristics of turbulence and dispersion. These range from the detailed examination of the production of turbulence kinetic energy, through examination of soundings plotted on thermo diagrams, to look-up tables such as those suggested by Pasquill and Gifford.

Finally, although we used the words “smoke” and “smoke stack” in this chapter, most emissions in North America and Europe are sufficiently clean that particulate matter is not visible. This clean-up has been expensive, but commendable.

A SCIENTIFIC PERSPECTIVE • Citizen Scientist

Scientists and engineers have at least the same responsibilities to society as do other citizens. Like our fellow citizens, we ultimately decide the short-term balance between environmental quality and material wealth, by the goods that we buy and by the government leaders we elect. Be informed. Take a stand. Vote.

Perhaps we have more responsibility, because we can also calculate the long-term consequences of our actions. We have the ability to evaluate various options and build the needed technology. Take action. Discover the facts. Design solutions.