2 Transport Processes

2.1 Diffusion Without Shear

(i) Instantaneous plane source in 3-dimensions

Suppose that the initial distribution of $C$ is dependent only on $x$ and $t$ and that $u$, $v$ and $w$ are constant. Then Eq. (14) reduces to

\begin{displaymath} {\partial C\over \partial t}+u{\partial C\over \partial x}= ... ...varepsilon _x{\partial C\over \partial x}\right ). \eqno{(15)} \end{displaymath}

We now transform to a frame of reference moving with the flow speed $u$:

\begin{displaymath} X=x-ut. \end{displaymath}

Then Eq. (15) becomes

\begin{displaymath} {\partial C\over \partial t}= {\partial\over \partial X}\lef... ...varepsilon _x{\partial C\over \partial X}\right ). \eqno{(16)} \end{displaymath}

Suppose that $\varepsilon _x$ is constant. Then the solution to Eq. (16) is

\begin{displaymath} C(x,t)={Q\over 2\sqrt{\pi\varepsilon _xt}}{\rm exp}\left (-{X^2\over 4\varepsilon _xt}\right ). \eqno{(17)} \end{displaymath}

It can be shown that

\begin{displaymath} \lim_{t\rightarrow 0}C(x,t)=Q\delta (X)=Q\delta (x).\eqno{(18)} \end{displaymath}

The solution (17) can be interpreted as an instantaneous release of an amount $Q$ kgm$^{-2}$ of pollutant at $t=0$ in the plane $X=x=0$. Eq. (17) is the Gaussian ``puff'' solution because it describes the release of a small ``puff'' of pollutant.

An important quantity is the mean square distance $\sigma_x^2$ which the pollutant has spread from $X=0$. This is given by

\begin{displaymath} \sigma_x^2={1\over Q}\int_{-\infty}^\infty CX^2\,{\rm d}X=2\varepsilon _xt.\eqno{(19)} \end{displaymath}

Therefore

\begin{displaymath} C(x,t)={Q\over \sigma_x\sqrt{2\pi}}{\rm exp}\left (-{X^2\over 2\sigma_x^2}\right ). \eqno{(20)} \end{displaymath}

The pollutant therefore spreads at a rate proportional to $\sqrt{t}$.

(ii) Instantaneous line source in 3-dimensions

In this case we assume that $C=C(x,y,t)$ so that the equation for $C$ is

\begin{displaymath} {\partial C\over \partial t}+u{\partial C\over \partial x} +... ...l y}\left (\varepsilon _y{\partial C\over \partial y}\right ). \end{displaymath}

As in the previous case we make the transformation $X=x-ut$ and also let $Y=y-vt$. Then

\begin{displaymath} {\partial C\over \partial t}= {\partial\over \partial X}\lef... ...varepsilon _y{\partial C\over \partial Y}\right ). \eqno{(21)} \end{displaymath}

The solution is

\begin{displaymath} C(X,Y,t)={Q\over 4\pi t\sqrt{\varepsilon _x\varepsilon _y}}{... ...ver \varepsilon _x}+{Y^2\over \varepsilon _y}\right )\right ]. \end{displaymath}

It can be shown that $\sigma_x^2=2\varepsilon _xt$ and $\sigma_y^2=2\varepsilon _yt$. Hence

\begin{displaymath} C(X,Y,t)={Q\over 2\pi\sigma_x\sigma_y}{\rm exp}\left [-{1\ov... ... \sigma_x^2}+{Y^2\over \sigma_y^2}\right )\right ].\eqno{(22)} \end{displaymath}

(iii) Instantaneous point source in 3-dimensions

This is the most general and practical case where $C=C(x,y,z,t)$. The equation for $C$ is

\begin{displaymath} {\partial C\over \partial t}+u{\partial C\over \partial x} +... ...l z}\left (\varepsilon _z{\partial C\over \partial z}\right ). \end{displaymath}

We could put $Z=z-wt$ in line with the cases of plane and line sources but it is difficult to think of real cases where there is a mean vertical velocity $w$ so we keep $w=0$. The transsformed equation for $C$ is then

\begin{displaymath} {\partial C\over \partial t}= {\partial\over \partial X}\lef... ...varepsilon _z{\partial C\over \partial z}\right ). \eqno{(23)} \end{displaymath}

The solution is

\begin{displaymath} C(X,Y,z,t)={Q\over 8(\pi t)^{3/2}\sqrt{\varepsilon _x\vareps... ...er \varepsilon _y}+ {z^2\over \varepsilon _z}\right )\right ]. \end{displaymath}

In this case $\sigma_x^2=2\varepsilon _xt$, $\sigma_y^2=2\varepsilon _yt$ and $\sigma_z^2=2\varepsilon _zt$ so that

\begin{displaymath} C(X,Y,z,t)={Q\over (2\pi)^{3/2}\sigma_x\sigma_y\sigma_z} {\r... ...\sigma_y^2}+{z^2\over \sigma_z^2}\right ) \right ].\eqno{(24)} \end{displaymath}

Fig. 2. Concentration distributions at various times after pollutant release for the diffusion of an instantaneous plane source (Eq. (20)).

2.2 Advection Without Diffusion in a Shear Flow

Suppose that we have a uni-directional shear flow:

\begin{displaymath} {\bf u}=(u,0,0)=\left ({Uz\over h},0,0\right )\eqno{(25)} \end{displaymath}

and suppose that we are interested in the flow between $z=0$ and $z=h$.

Fig. 3. Shear flow $u=Uz/h$.

If we neglect diffusion, then the equation for $C$ is

\begin{displaymath} {\partial C\over \partial t}+u{\partial C\over \partial x}=0.\eqno{(26)} \end{displaymath}

If we again put $X=x-u(z)t$ Eq. (26) becomes

\begin{displaymath} {\partial C(X,t)\over \partial t}=0\eqno{(27)} \end{displaymath}

and so $C=C(X)$, i.e. the concentration is independent of time moving with the local flow $u(z)$. Consider as in the previous section the case of an instantaneous plane source at $x=0$, $t=0$. Then at $t=0$

\begin{displaymath} C(X)=Q\delta (X). \end{displaymath}

This must be the solution for all time, because $\partial C/\partial t=0$. Therefore

\begin{displaymath} C=Q\delta (x-u(z)t)=Q\delta\left (x-{Uz\over h}\right ).\eqno{(28)} \end{displaymath}

Fig. 4. A line (or plane in 3-D) of pollutant initially lying on $x=0$ is stretched and rotated by a shear flow $u=Uz/h$. At time $t$ the $x$ extent of the line of pollutant is denoted by $l_x$.

Consider the horizontal extent $l_x$ of the pollutant. At time $t$ it is clearly

\begin{displaymath} l_x=Ut. \end{displaymath}

We can see that the pollutant spreads out at a rate proportional to $t$. This contrasts with the rate proportional to $\sqrt{t}$ for diffusion, so it would appear that advection is always more important than diffusion, except perhaps for a short time after release. However, we shall see in the next section that this is not the case.

2.3 Advection and Diffusion in a Shear Flow

When advection and diffusion occur simultaneously, the stretching of regions of high concentration, as occurs for pure advection, is less effective because transverse diffusion of the high concentration region is enhanced by the stretching caused by the shear. This is shown schematically in Fig. 5.

Fig. 5. An initial slab (a) of pollutant is rotated and stretched by shear (b), leading to enhanced diffusion in the transverse direction (c). The spreading of the line of maximum concentration is clearly reduced by the diffusion.

The processes described here are known as Taylor's mechanism.

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