6. Gaussian Plumes from High Chimneys

The Gaussian plume equation (47) assumed that the source was at $x=y=z=0$. We can easily change this so that the source is from a chimney of height $h$ by replacing Eq. (47) by

\begin{displaymath} C(x,y,z)={q\over 2\pi\sigma_y\sigma_zu}{\rm exp}\left ( -{y^... ...er 2\sigma_y^2}-{(z-h)^2\over 2\sigma_z^2}\right ).\eqno{(58)} \end{displaymath}

However, this equation tells us that diffusion takes place not only for $z>0$ (above the ground) but also for $z<0$ (below the ground). The simplest assumption available to correct this is to assume that any pollutant which reaches the ground is reflected back. Thus the pollutant which, according to Eq. (58) should be below the ground actually appears above the ground. This is exactly equivalent to allowing Eq. (58) to apply unmodified but then introducing another source at $z=-h$ (i.e. the image of the true source in the ground). We then replace Eq. (58) by

\begin{displaymath} C(x,y,z)={q\over 2\pi\sigma_y\sigma_zu}\left [{\rm exp}\left... ...ma_y^2}-{(z+h)^2\over 2\sigma_z^2}\right ) \right ]\eqno{(59)} \end{displaymath}

and use Eq. (59) for $z>0$ only. The concentration at ground level is

\begin{displaymath} C(x,y)={q\over \pi\sigma_y\sigma_zu}{\rm exp}\left (-{y^2\over 2\sigma_y^2}- {h^2\over 2\sigma_z^2}\right ).\eqno{(60)} \end{displaymath}

Fig. 12 shows the concentration at ground level downwind of two chimneys (one 25 m high and the other 50 m high) emitting the same amount of pollutant. It is clear that for the higher chimney the maximum concentration is both lower and further down wind.

Fig. 12. The ground level concentration of pollutant down wind of two chimneys emitting equal amounts of pollutant. Pasquill stability class C is assumed.

If we consider the Pasquill-Gifford stability class C (slightly unstable) then $\sigma_y\approx 100x^{0.9}$ and $\sigma_z\approx 60x^{0.9}$. The maximum ground level concentration according to Eq. (60) can then be shown to be at $\sigma_z=h/\sqrt{2}$ and its value is

\begin{displaymath} C_{\rm max}\approx{q\over 0.83\,\pi euh^2}\approx {0.15q\over h^2}.\eqno{(61)} \end{displaymath}

Note that the maximum concentration decreases as the square of the chimney height -- so tall chimneys are a good idea.

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