I
Flow lines in Atmospheric Diffusion
IN
A sense, flux of matter in turbulent diffusion is a flow phenomenon and, therefore, the question can be asked: How far can concepts of streamlines and equipotential lines be applied? Generally, there should be no difficulty in defining that which is equivalent to streamlines-e.g., matter discharged from a source such as a chimney must go somewhere and its path can usually be followed. General methods for attacking such problems are available in texts on theoretical physics (2) and these were applied to turbulent diffusion from a continuous point source into a steady wind. In a previous analysis, Batchelor (7) shows that net transport of matter in the diffusing field (infinite medium) can be represented by a transport or flux vector having the components,
T , = Uc
I n T,, a term containing &/ax is omitted because of an approximation usual in dealing with the continuous point source-namely, neglect of diffusion along wind. For spatial distribution of concentration, Sutton's equation (3) may be used in a slightly generalized form:
where S, and S, (standard deviations) may be arbitrary functions of x . In the illustrations, a form of S, and S, has been used which is based on an exponential decay-type Lagrangian correlation coefficient and is fitted to Sutton's power-law formula. Curves which have the transport vector direction at all points must satisfy the differential equation,
b e h e e n them are obtained by assigning a series of values to F / Q with. equal increments, and calculating the corresponding C1 - s from Equation 5 (Figure 1). By definition, pseudopotential surfaces are those everywhere normal to the streamlines. Their differential equation is
(3)
These are relatively unimportant configurations, but to quote some results, near x = 0, the pseudopotential surfaces become cylinders with their axes along the x- axis. O n the other hand, for large ./.YO
d x / T Z = d y / T y = dz/T,
Combining Equations 1, 2, and 3, the solution is: y = CIS,
2
= CZS,
(4)
These equations specify one streamlineeg., if C2 = 0, streamlines in a horizontal plane through the source are obtained. A slightly different interpretation of Equation 4 is that by Equation 2 where a fixed amount of matter flows between the xz plane and a surface characterized by y/S, = const. so that flow is along this surface. A similar conclusion applies to surfaces z/Sz = const., the intersection of two surfaces characterizing a streamline -one, y = const. S, and the other, z = const. S,. The fraction of matter between the two stream surfaces, y = - C1 S, and y = f C1 S, with z extending, for example, to infinity in both directions is given by F / Q = erf(CI/&)
(5)
Then, in a horizontal slice through the source, the streamlines for equal flux
T , dx f T , dy
x2
- 2xxo
+ T , dz
+ y2/2 + z2/2
= 0
=
(6)
c
(7)
That is, ellipsoids centered at x = X O , y = 0, z = 0, the principal axis along x being 0.707 times greater than that along y or z. Surfaces of equal concentration may be constructed by solving the following differential equation
eax & +by - &aca2-+d z = 0 ac
(8)
Assuming that scales of diffusion are identical in the y and z directions-i.e., S, = uS$-the variables are separable and Equation 8 reduces to r2 =
c2
4 s,2 log (C/S,)
+
(9)
where =y2 u2z2. With S, given as a function of x, these surfaces (of parameter C) may be constructed as shown in Figure 2 which VOL. 49, NO. 9
SEPTEMBER 1957
1453
Figure 1.
Figure 2.
Diffusion from a continuous point source; streamlines in plan have a 21/270 total flux between neighboring lines
lsotimic surfaces cut b y horizontal plane through the source;
again is a horizontal section through the source, illustrates the same range in diffusion as Figure 1. To connect parameter C to the (constant) concentration on the surface it is best to consider the point S, = C, = 0 :
So that for a series of isotimic surfaces on which the concentration reduces by a constant factor, rn, value of the parameter increases with the square root of m. Isotimic lines on ground level may also be easily constructed. The constant,
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concentration decreases b y a factor of 2 between lines
apparently has the effect of increasing chimney height by a factor of a.
a,
Nomenclature x,
y, z
=
T,: Tu,T ,
=
S,,Sz
=
L'
=
c
Q
= = =
F
=
C, CI, Cz
INDUSTRIAL AND ENGINEERING CHEMISTRY
space cosrdinates along wind, across wind, and in the vertical, respectively components of transport vector standard deviations in the horizontal and vertical mean wind speed concentration of pollutant rate of emission by source surface parameters flow rate of pollutant
.YO
= scale of diffusion: relaxation
length of Lagrangian correlation coefficient literature Cited
(1) Batchelor, G. K., Australian J . Sci. Research 2, 437-50 (1949). ( 2 ) Morse. P. M., Feshbach, H., "Methods of Theoretical Physics." McGrawHill, New York. 1953. (3) Sutton, 0. G.. "Micrometeorology," McGraw-Hill, New York, 1953.
G. T. CSANADY Wollongong College, New South Wales University of Technology, New South Wales, Australia
